problem solving questions for math

• Top Questions

Convert the following readings of pressureto kPa absolute, assuming that the barometer reads 760mm Hg:...

Show that an element and its inverse have the same order in any group.

A cubic block of wood, 10.0 cm on each side, floats at the interface between...

An ice cube tray of negligible mass contains 0.315 kg of water at17.7∘. How much...

An open tank has a vertical partition and on one side contains gasoline with a...

A 100g cube of ice at 0C is dropped into 1.0kg of water thatwas originally...

Complete the equation of the line through (2, 1) and (5, -8). Use exact numbers.

Write a function based on the given parent function and transformations in the given order....

How many solutions does the equationx1+x2+x3=13have wherex1,x2,andx3are non negative integers less than 6.

Determine whether f : Z×Z→Z is onto if a) f(m,n)=2m−nb) b) f(m,n)=m2−n2 c) f(m,n)=m+n+1 d)...

The base of S is an elliptical region with boundary curve 9x2+4y2=36. Cross-sections perpendicular to...

Create a graph of y=2x−6. Construct a graph corresponding to the linear equation y=2x−6.

Use the graphs of f and g to graph h(x) = (f + g)(x). (Graph...

find expressions for the quadratic functions whose graphs are shown. f(x)=? g(x)=?

Find the volume V of the described solid S. A cap of a sphere with...

Find a counterexample to show that each statement is false. The sum of any three...

Read the numbers and decide what the next number should be. 5 15 6 18...

In how many different orders can five runners finish a race if no ties are...

A farmer plants corn and wheat on a 180 acre farm. He wants to plant...

Find the distance between (0, 0) and (-3, 4) pair of points. If needed, show...

Whether each of these functions is a bijection from R to R.a)f(x)=−3x+4b)f(x)=−3x2+7c)f(x)=x+1x+2d)f(x)=x5+1

Find two numbers whose difference is 100 and whose product is a minimum.

Find an expression for the function whose graph is the given curve. The line segment...

Prove or disprove that if a and b are rational numbers, then ab is also...

How to find a rational number halfway between any two rational numbers given infraction form...

Fill in the blank with a number to make the expression a perfect square x2−6x+?

Look at this table: x y 1–2 2–4 3–8 4–16 5–32 Write a linear (y=mx+b),...

Part a: Assume that the height of your cylinder is 8 inches. Think of A...

The graph of a function f is shown. Which graph is an antiderivative of f?

A rectangle has area 16m2 . Express the perimeter of the rectangle as a function...

Find the equation of the quadratic function f whose graph is shown below. (5, −2)

Use the discriminant, b2−4ac, to determine the number of solutions of the following quadratic equation....

How many solutions does the equation ||2x-3|-m|=m have if m>0?

If a system of linear equations has infinitely many solutions, then the system is called...

A bacteria population is growing exponentially with a growth factor of 16 each hour.By what...

A system of linear equations with more equations than unknowns is sometimes called an overdetermined...

Express the distance between the numbers 2 and 17 using absolute value. Then find the...

Find the Laplace transform of f(t)=(sin⁡t–cos⁡t)2

Express the interval in terms of an inequality involving absolute value. (0,4)

A function is a ratio of quadratic functions and has a vertical asymptote x =4...

how do you graph y > -2

Find the weighted average of a data set where 10 has a weight of 5,...

The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume...

The population of a region is growing exponentially. There were 10 million people in 1980...

Two cables BG and BH are attached to the frame ACD as shown.Knowing that the...

A bird flies in the xy-plane with a position vector given by r→=(αt−βt3)i^+γt2j^, with α=2.4...

A movie stuntman (mass 80.0kg) stands on a window ledge 5.0 mabove the floor. Grabbing...

Solve the following linear congruence, 25x≡15(bmod29)

For the equation, a. Write the value or values of the variable that make a...

Which of the following statements is/are correct about logistic regression? (There may be more than...

Compute 4.659×104−2.14×104. Round the answer appropriately. Express your answer as an integer using the proper...

Find the 52nd term of the arithmetic sequence -24,-7, 10

Find the 97th term of the arithmetic sequence 17, 26, 35,

An equation that expresses a relationship between two or more variables, such as H=910(220−a), is...

The football field is rectangular. a. Write a polynomial that represents the area of the...

The equation 1.5r+15=2.25r represents the number r of movies you must rent to spend the...

While standing on a ladder, you drop a paintbrush. The function represents the height y...

When does data modeling use the idea of a weak entity? Give definitions for the...

Write an equation of the line passing through (-2, 5) and parallel to the line...

Find a polar equation for the curve represented by the given Cartesian equation. y =...

Find c such that fave=f(c)

List five integers that are congruent to 4 modulo 12.

A rectangular package to be sent by a postal service can have a maximum combined...

A juggler throws a bowling pin straight up with an initial speed of 8.20 m/s....

The One-to-One Property of natural logarithms states that if ln x = ln y, then...

Find an equation of a parabola that has curvature 4 at the origin.

Find a parametric representation of the solution set of the linear equation. 3x − 1/2y...

Find the product of the complex number and its conjugate. 2-3i

Find the prime factorization of 10!.

Find a polynomial f(x) of degree 5 that has the following zeros. -3, -7, 5...

True or False. The domain of every rational function is the set of all real...

What would be the most efficient step to suggest to a student attempting to complete...

Write a polynomial, P(x), in factored form given the following requirements. Degree: 4 Leading coefficient...

Give a geometric description of the set of points in space whose coordinates satisfy the...

Use the Cauchy-Riemann equations to show that f(z)=z― is not analytic.

Find the local maximum and minimum values and saddle points of the function. If you...

a) Evaluate the polynomial y=x3−7x2+8x−0.35 at x=1.37 . Use 3-digit arithmetic with chopping. Evaluate the...

The limit represents f'(c) for a function f and a number c. Find f and...

A man 6 feet tall walks at a rate of 5 feet per second away...

Find the Maclaurin series for the function f(x)=cos⁡4x. Use the table of power series for...

Suppose that a population develops according to the logistic equation dPdt=0.05P−0.0005P2 where t is measured...

Find transient terms in this general solution to a differential equation, if there are any...

Find the lengths of the sides of the triangle PQR. Is it a right triangle?...

Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -4),...

Find a path that traces the circle in the plane y=5 with radius r=2 and...

a. Find an upper bound for the remainder in terms of n.b. Find how many...

Find two unit vectors orthogonal to both j−k and i+j.

Obtain the Differential equations: parabolas with vertex and focus on the x-axis.

The amount of time, in minutes, for an airplane to obtain clearance for take off...

Use the row of numbers shown below to generate 12 random numbers between 01 and...

Here’s an interesting challenge you can give to a friend. Hold a $1 (or larger!)...

A random sample of 1200 U.S. college students was asked, "What is your perception of...

The two intervals (114.4, 115.6) and (114.1, 115.9) are confidence intervals (The same sample data...

How many different 10 letter words (real or imaginary) can be formed from the following...

Assume that σ is unknown, the lower 100(1−α)% confidence bound on μ is: a) μ≤x―+tα,n−1sn...

A simple random sample of 60 items resulted in a sample mean of 80. The...

Decresing the sample size, while holding the confidence level and the variance the same, will...

A privately owned liquor store operates both a drive-n facility and a walk-in facility. On...

Show that the equation represents a sphere, and find its center and radius. x2+y2+z2+8x−6y+2z+17=0

Describe in words the region of R3 represented by the equation(s) or inequality. x=5

Suppose that the height, in inches, of a 25-year-old man is a normal random variable...

Find the value and interest earned if $8906.54 is invested for 9 years at %...

Which of the following statements about the sampling distribution of the sample mean is incorrect?...

The random variable x stands for the number of girls in a family of four...

The product of the ages, in years, of three (3) teenagers os 4590. None of...

A simple random sample size of 100 is selected from a population with p=0.40 What...

Which of the following statistics are unbiased estimators of population parameters? Choose the correct answer...

The probability distribution of the random variable X represents the number of hits a baseball...

Let X be a random variable with probability density function.f(x)={c(1−x2)−1<x<10otherwise(a) What is the value of...

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do...

The monthly worldwide average number of airplane crashes of commercial airlines is 2.2. What is...

Given that z is a standard normal random variable, compute the following probabilities.a.P(z≤−1.0)b.P(z≥−1)c.P(z≥−1.5)d.P(−2.5≤z)e.P(−3<z≤0)

Given a standard normal distribution, find the area under the curve that lies(a) to the...

Chi-square tests are best used for which type of dependent variable? nominal, ordinal ordinal interval...

True or False 1.The goal of descriptive statistics is to simplify, summarize, and organize data....

What is the difference between probability distribution and sampling distribution?

A weather forecaster predicts that the temperature in Antarctica will decrease 8∘F each hour for...

The tallest person who ever lived was approximately 8 feet 11 inches tall. a) Write...

An in-ground pond has the shape of a rectangular prism. The pond has a depth...

The average zinc concentration recovered from a sample of measurements taken in 36 different locations...

Why is it important that a sample be random and representative when conducting hypothesis testing?...

Which of the following is true about the sampling distribution of means? A. Shape of...

Give an example of a commutative ring without zero-divisors that is not an integral domain.

List all zero-divisors in Z20. Can you see relationship between the zero-divisors of Z20 and...

Find the integer a such that a≡−15(mod27) and −26≤a≤0

Explain why the function is discontinuous at the given number a. Sketch the graph of...

Two runners start a race at the same time and finish in a tie. Prove...

Which of the following graphs represent functions that have inverse functions?

find the Laplace transform of f (t). f(t)=tsin⁡3t

Find Laplace transforms of sin⁡h3t cos22t

find the Laplace transform of f (t). f(t)=t2cos⁡2t

The Laplace transform of the product of two functions is the product of the Laplace...

The Laplace transform of u(t−2) is (a) 1s+2 (b) 1s−2 (c) e2ss(d)e−2ss

Find the Laplace Transform of the function f(t)=eat

Explain First Shift Theorem & its properties?

Solve f(t)=etcos⁡t

Find Laplace transform of the given function te−4tsin⁡3t

Reduce to first order and solve:x2y″−5xy′+9y=0 y1=x3

(D3−14D+8)y=0

A thermometer is taken from an inside room to the outside ,where the air temperature...

Find that solution of y′=2(2x−y) which passes through the point (0, 1).

Radium decomposes at a rate proportional to the amount present. In 100 years, 100 mg...

Let A, B, and C be sets. Show that (A−B)−C=(A−C)−(B−C)

Suppose that A is the set of sophomores at your school and B is the...

In how many ways can a 10-question true-false exam be answered? (Assume that no questions...

Is 2∈{2}?

How many elements are in the set { 2,2,2,2 } ?

How many elements are in the set { 0, { { 0 } }?

Draw the Hasse diagram representing the partial ordering {(a, b) | a divides b} on...

Flux through a Cube (Eigure 1) A cube has one corner at the origin and...

A well-insulated rigid tank contains 3 kg of saturated liquid-vapor mixture of water at 200...

A water pump that consumes 2 kW of electric power when operating is claimed to...

A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius...

In a truck-loading station at a post office, a small 0.200-kg package is released from...

The magnetic fieldB→in acertain region is 0.128 ,and its direction is that of the z-axis...

A marble moves along the x-axis. The potential-energy functionis shown in Fig. 1a) At which...

A proton is released in a uniform electric field, and it experiences an electric force...

A potters wheel having a radius of 0.50 m and a moment of inertia of12kg⋅m2is...

Two spherical objects are separated by a distance of 1.80×10−3m. The objects are initially electrically...

An airplane pilot sets a compass course due west and maintainsan airspeed of 220 km/h....

Resolve the force F2 into components acting along the u and v axes and determine...

A conducting sphere of radius 0.01m has a charge of1.0×10−9Cdeposited on it. The magnitude of...

Starting with an initial speed of 5.00 m/s at a height of 0.300 m, a...

In the figure a worker lifts a weightωby pulling down on a rope with a...

A stream of water strikes a stationary turbine bladehorizontally, as the drawing illustrates. The incident...

Until he was in his seventies, Henri LaMothe excited audiences by belly-flopping from a height...

A radar station, located at the origin of xz plane, as shown in the figure...

Two snowcats tow a housing unit to a new location at McMurdo Base, Antarctica, as...

You are on the roof of the physics building, 46.0 m above the ground. Your...

A block is on a frictionless table, on earth. The block accelerates at5.3ms2when a 10...

A 0.450 kg ice puck, moving east with a speed of3.00mshas a head in collision...

A uniform plank of length 2.00 m and mass 30.0 kg is supported by three...

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope...

A ski tow operates on a 15.0 degrees slope of lenth 300m. The rope moves...

Two blocks with masses 4.00 kg and 8.00 kg are connected by string and slide...

From her bedroom window a girl drops a water-filled balloon to the ground 6.0 m...

A 730-N man stands in the middle of a frozen pond of radius 5.0 m....

A 5.00 kg package slides 1.50 m down a long ramp that is inclined at12.0∘below...

Ropes 3m and 5m in length are fastened to a holiday decoration that is suspended...

A skier of mass 70 kg is pulled up a slope by a motor driven...

A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid,...

A sled with rider having a combined mass of 120 kg travels over the perfectly...

A 7.00- kg bowling ball moves at 3.00 m/s. How fast must a 2.45- g...

Two point chargesq1=+2.40nC andq2=−6.50nC are 0.100 m apart. Point A is midway between them and...

A block of mass m slides on a horizontal frictionless table with an initial speed...

A space traveler weights 540 N on earth. what will the traveler weigh on another...

A block of mass m=2.20 kg slides down a 30 degree incline which is 3.60...

A weatherman carried an aneroid barometer from the groundfloor to his office atop a tower....

If a negative charge is initially at rest in an electric field, will it move...

A coin with a diameter of 2.40cm is dropped on edge on to a horizontal...

An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction...

An 80.0-kg skydiver jumps out of a balloon at an altitude of1000 m and opens...

A 0.145 kg baseball pitched at 39.0 m/s is hit on a horizontal line drive...

A 1000 kg safe is 2.0 m above a heavy-duty spring when the rope holding...

A 500 g ball swings in a vertical circle at the end of a1.5-m-long string....

A rifle with a weight of 30 N fires a 5.0 g bullet with a...

The tires of a car make 65 revolutions as the car reduces its speed uniformly...

A 2.0- kg piece of wood slides on the surface. The curved sides are perfectly...

A 292 kg motorcycle is accelerating up along a ramp that is inclined 30.0° above...

A projectile is shot from the edge of a cliff 125 m above ground level...

A lunch tray is being held in one hand, as the drawing illustrates. The mass...

The initial velocity of a car, vi, is 45 km/h in the positivex direction. The...

An Alaskan rescue plane drops a package of emergency rations to a stranded party of...

Raindrops make an angle theta with the vertical when viewed through a moving train window....

A 0.50 kg ball that is tied to the end of a 1.1 m light...

If the coefficient of static friction between your coffeecup and the horizontal dashboard of your...

A car is initially going 50 ft/sec brakes at a constant rate (constant negative acceleration),...

A swimmer is capable of swimming 0.45m/s in still water (a) If sheaim her body...

A block is hung by a string from inside the roof of avan. When the...

A race driver has made a pit stop to refuel. Afterrefueling, he leaves the pit...

A relief airplane is delivering a food package to a group of people stranded on...

The eye of a hurricane passes over Grand Bahama Island. It is moving in a...

An extreme skier, starting from rest, coasts down a mountainthat makes an angle25.0∘with the horizontal....

Four point charges form a square with sides of length d, as shown in the...

In a scene in an action movie, a stuntman jumps from the top of one...

The spring in the figure (a) is compressed by length delta x . It launches...

An airplane propeller is 2.08 m in length (from tip to tip) and has a...

A helicopter carrying dr. evil takes off with a constant upward acceleration of5.0ms2. Secret agent...

A 15.0 kg block is dragged over a rough, horizontal surface by a70.0 N force...

A box is sliding with a speed of 4.50 m/s on a horizontal surface when,...

3.19 Win the Prize. In a carnival booth, you can win a stuffed giraffe if...

A car is stopped at a traffic light. It then travels along a straight road...

a. When the displacement of a mass on a spring is12A, what fraction of the...

At a certain location, wind is blowing steadily at 10 m/s. Determine the mechanical energy...

A jet plane lands with a speed of 100 m/s and can accelerate at a...

In getting ready to slam-dunk the ball, a basketball player starts from rest and sprints...

An antelope moving with constant acceleration covers the distance between two points 70.0 m apart...

A bicycle with 0.80-m-diameter tires is coasting on a level road at 5.6 m/s. A...

The rope and pulley have negligible mass, and the pulley is frictionless. The coefficient of...

A proton with an initial speed of 800,000 m/s is brought to rest by an...

The volume of a cube is increasing at the rate of 1200 cm supmin at...

An airplane starting from airport A flies 300 km east, then 350 km at 30...

To prove: In the following figure, triangles ABC and ADC are congruent. Given: Figure is...

Conduct a formal proof to prove that the diagonals of an isosceles trapezoid are congruent....

The distance between the centers of two circles C1 and C2 is equal to 10...

Segment BC is Tangent to Circle A at Point B. What is the length of...

Find an equation for the surface obtained by rotating the parabola y=x2 about the y-axis.

Find the area of the parallelogram with vertices A(-3, 0), B(-1 , 3), C(5, 2),...

If the atomic radius of lead is 0.175 nm, find the volume of its unit...

At one point in a pipeline the water’s speed is 3.00 m/s and the gauge...

Find the volume of the solid in the first octant bounded by the coordinate planes,...

A paper cup has the shape of a cone with height 10 cm and radius...

A light wave has a 670 nm wavelength in air. Its wavelength in a transparent...

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50...

Find the equation of the sphere centered at (-9, 3, 9) with radius 5. Give...

Determine whether the congruence is true or false. 5≡8 mod 3

Find all whole number solutions of the congruence equation. (2x+1)≡5 mod 4

Determine whether the congruence is true or false. 100≡20 mod 8

I want example of an undefined term and a defined term in geometry and explaining...

Two fair dice are rolled. Let X equal the product of the 2dice. Compute P{X=i}...

Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The...

Based on the Normal model N(100, 16) describing IQ scores, what percent of peoples

The probability density function of the net weight in pounds of a packaged chemical herbicide...

Let X represent the difference between the number of heads and the number of tails...

An urn contains 3 red and 7 black balls. Players A and B withdraw balls...

80% A poll is given, showing are in favor of a new building project. 8...

The probability that the San Jose Sharks will win any given game is 0.3694 based...

Find the value of P(X=7) if X is a binomial random variable with n=8 and...

Find the value of P(X=8) if X is a binomial random variable with n=12 and...

On a 8 question multiple-choice test, where each question has 2 answers, what would be...

If you toss a fair coin 11 times, what is the probability of getting all...

A coffee connoisseur claims that he can distinguish between a cup of instant coffee and...

Two firms V and W consider bidding on a road-building job, which may or may...

Two cards are drawn without replacement from an ordinary deck, find the probability that the...

In August 2012, tropical storm Isaac formed in the Caribbean and was headed for the...

A local bank reviewed its credit card policy with the intention of recalling some of...

The accompanying table gives information on the type of coffee selected by someone purchasing a...

A batch of 500 containers for frozen orange juice contains 5 that are defective. Two...

The probability that an automobile being filled with gasoline also needs an oil change is...

Let the random variable X follow a normal distribution with μ=80 and σ2=100. a. Find...

A card is drawn randomly from a standard 52-card deck. Find the probability of the...

The next number in the series 38, 36, 30, 28, 22 is ?

What is the coefficient of x8y9 in the expansion of (3x+2y)17?

A boat on the ocean is 4 mi from the nearest point on a straight...

How many different ways can you make change for a quarter? (Different arrangements of the...

Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue, and...

Approximately 80,000 marriages took place in the state of New York last year. Estimate the...

The probability that a student passes the Probability and Statistics exam is 0.7. (i)Find the...

Customers at a gas station pay with a credit card (A), debit card (B), or...

It is conjectured that an impurity exists in 30% of all drinking wells in a...

Assume that the duration of human pregnancies can be described by a Normal model with...

According to a renowned expert, heavy smokers make up 70% of lung cancer patients. If...

Two cards are drawn successively and without replacement from an ordinary deck of playing cards...

Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L),...

A bag contains 6 red, 4 blue and 8 green marbles. How many marbles of...

A normal distribution has a mean of 50 and a standard deviation of 4. Please...

Seven women and nine men are on the faculty in the mathematics department at a...

An automatic machine in a manufacturing process is operating properly if the lengths of an...

Three cards are drawn without replacement from the 12 face cards (jacks, queens, and kings)...

Among 157 African-American men, the mean systolic blood pressure was 146 mm Hg with a...

A TIRE MANUFACTURER WANTS TO DETERMINE THE INNER DIAMETER OF A CERTAIN GRADE OF TIRE....

Differentiate the three measures of central tendency: ungrouped data.

Find the mean of the following data: 12,10,15,10,16,12,10,15,15,13

A wallet containing four P100 bills, two P200 bills, three P500 bills, and one P1,000...

The number of hours per week that the television is turned on is determined for...

Data was collected for 259 randomly selected 10 minute intervals. For each ten-minute interval, the...

Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in...

A normal distribution has a mean of 80 and a standard deviation of 14. Determine...

True or false: a. All normal distributions are symmetrical b. All normal distributions have a...

Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or...

Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8408 1374 1872 8879...

The velocity function (in meters per second) is given for a particle moving along a...

Find the area of the parallelogram with vertices A(-3,0) , B(-1,6) , C(8,5) and D(6,-1)

What is the area of the parallelogram with vertices A(-3, 0), B(-1, 5), C(7, 4),...

The integral represents the volume of a solid. Describe the solid. π∫01(y4−y8)dy a) The integral...

Two components of a minicomputer have the following joint pdf for their useful lifetimes X...

Use the table of values of f(x,y) to estimate the values of fx(3,2), fx(3,2.2), and...

Calculate net price factor and net price. Dollars list price −435.20$ Trade discount rate −26%,15%,5%.

Represent the line segment from P to Q by a vector-valued function and by a...

(x2+2xy−4y2)dx−(x2−8xy−4y2)dy=0

If f is continuous and integral 0 to 9 f(x)dx=4, find integral 0 to 3...

Find the parametric equation of the line through a parallel to ba=[3−4],b=[−78]

Find the velocity and position vectors of a particle that has the given acceleration and...

If we know that the f is continuous and integral 0 to 4f(x)dx=10, compute the...

Integration of (y⋅tan⁡xy)

For the matrix A below, find a nonzero vector in the null space of A...

Find a nonzero vector orthogonal to the plane through the points P, Q, and R....

Suppose that the augmented matrix for a system of linear equations has been reduced by...

Find two unit vectors orthogonal to both (3 , 2, 1) and (- 1, 1,...

What is the area of the parallelogram whose vertices are listed? (0,0), (5,2), (6,4), (11,6)

Using T defined by T(x)=Ax, find a vector x whose image under T is b,...

Use the definition of Ax to write the matrix equation as a vector equation, or...

We need to find the volume of the parallelepiped with only one vertex at the...

List five vectors in Span {v1,v2}. For each vector, show the weights on v1 and...

(1) find the projection of u onto v and (2) find the vector component of...

Find the area of the parallelogram determined by the given vectors u and v. u...

(a) Find the point at which the given lines intersect. r = 2,...

(a) find the transition matrix from B toB′,(b) find the transition matrix fromB′to B,(c) verify...

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If...

Given the following vector X, find anon zero square marix A such that AX=0; You...

Construct a matrix whose column space contains (1, 1, 5) and (0, 3.1) and whose...

At what point on the paraboloid y=x2+z2 is the tangent plane parallel to the plane...

Label the following statements as being true or false. (a) If V is a vector...

Find the Euclidean distance between u and v and the cosine of the angle between...

Write an equation of the line that passes through (3, 1) and (0, 10)

There are 100 two-bedroom apartments in the apartment building Lynbrook West.. The montly profit (in...

State and prove the linearity property of the Laplace transform by using the definition of...

The analysis of shafts for a compressor is summarized by conformance to specifications. Suppose that...

The Munchies Cereal Company combines a number of components to create a cereal. Oats and...

Movement of a Pendulum A pendulum swings through an angle of 20∘ each second. If...

If sin⁡x+sin⁡y=aandcos⁡x+cos⁡y=b then find tan⁡(x−y2)

Find the values of x such that the angle between the vectors (2, 1, -1),...

Find the dimensions of the isosceles triangle of largest area that can be inscribed in...

Suppose that you are headed toward a plateau 50 meters high. If the angle of...

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport....

Find an equation of the plane. The plane through the points (2, 1, 2), (3,...

Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following...

two small spheres spaced 20.0cm apart have equal charges. How many extra electrons must be...

The base of a pyramid covers an area of 13.0 acres (1 acre =43,560 ft2)...

Find out these functions' domain and range. To find the domain in each scenario, identify...

Your bank account pays an interest rate of 8 percent. You are considering buying a...

Whether f is a function from Z to R ifa)f(n)=±n.b)f(n)=n2+1.c)f(n)=1n2−4.

The probability density function of X, the lifetime of a certain type of electronic device...

A sandbag is released by a balloon that is rising vertically at a speed of...

A proton is located in a uniform electric field of2.75×103NCFind:a) the magnitude of the electric...

A rectangular plot of farmland are finite on one facet by a watercourse and on...

A solenoid is designed to produce a magnetic field of 0.0270 T at its center....

I want to find the volume of the solid enclosed by the paraboloidz=2+x2+(y−2)2and the planesz=1,x=−1y=0,andy=4

Let W be the subspace spanned by the u’s, and write y as the sum...

Can u find the point on the planex+2y+3z=13that is closest to the point (1,1,1). You...

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force...

A force of 250 Newtons is applied to a hydraulic jack piston that is 0.01...

Three identical blocks connected by ideal strings are being pulled along a horizontal frictionless surface...

A credit card contains 16 digits between 0 and 9. However, only 100 million numbers...

Every real number is also a complex number? True of false?

Let F be a fixed 3x2 matrix, and let H be the set of all...

Find a vector a with representation given by the directed line segment AB. Draw AB...

Find A such that the given set is Col A. {[2s+3tr+s−2t4r+s3r−s−t]:r,s,t real}

Find the vector that has the same direction as (6, 2, -3) but is four...

For the matrices (a) find k such that Nul A is a subspace of Rk,...

How many subsets with an odd number of elements does a set with 10 elements...

In how many ways can a set of five letters be selected from the English...

Suppose that f(x) = x/8 for 3 < x < 5. Determine the following probabilities:...

Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to...

Find two vectors parallel to v of the given length. v=PQ→ with P(1,7,1) and Q(0,2,5);...

A dog in an open field runs 12.0 m east and then 28.0 m in...

Can two events with nonzero probabilities be both independent and mutually exclusive? Explain your reasoning.

Use the Intermediate Value Theorem to show that there is a root of the given...

In a fuel economy study, each of 3 race cars is tested using 5 different...

A company has 34 salespeople. A board member at the company asks for a list...

A dresser drawer contains one pair of socks with each of the following colors: blue,...

A restaurant offers a $12 dinner special with seven appetizer options, 12 choices for an...

A professor writes 40 discrete mathematics true/false questions. Of the statements in these questions, 17...

Suppose E(X)=5 and E[X(X–1)]=27.5, find ∈(x2) and the variance.

A Major League baseball diamond has four bases forming a square whose sides measure 90...

Express f(x)=4x3+6x2+7x+2 in term of Legendre Polynomials.

Find a basis for the space of 2×2 diagonal matrices. Basis ={[],[]}

Which of the following expressions are meaningful? Which are meaningless? Explain. a) (a⋅b)⋅c (a⋅b)⋅c has...

Vectors V1 and V2 are different vectors with lengths V1 and V2 respectively. Find the...

Find an equation for the plane containing the two (parallel) lines v1=(0,1,−2)+t(2,3,−1) and v2=(2,−1,0)+t(2,3,−1).

Find, correct to the nearest degree, the three angles of the triangle with the given...

Find the vector, not with determinants, but by using properties of cross products. (i+j)×(i−j)

Find the curve’s unit tangent vector. Also, find the length of the indicated portion of...

Construct a 4×3 matrix with rank 1

Find x such that the matrix is equal to its inverse.A=[7x−8−7]

Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3...

Write in words how to read each of the following out loud.a.{x∈R′∣0<x<1}b.{x∈R∣x≤0orx⇒1}c.{n∈Z∣nisafactorof6}d.{n∈Z⋅∣nisafactorof6}

Pets Plus and Pet Planet are having a sale on the same aquarium. At Pets...

Find the average value of F(x, y, z) over the given region. F(x,y,z)=x2+9 over the...

Find the trace of the plane in the given coordinate plane. 3x−9y+4z=5,yz

Determine the level of measurement of the variable. Favorite color Choose the correct level of...

How wide is the chasm between what men and women earn in the workplace? According...

Write an algebraic expression for: 6 more than a number c.

Please, can u convert 3.16 (6 repeating) to a fraction.

Evaluate the expression. P(8, 3)

In a poker hand consisting of 5 cards, find the probability of holding 3 aces.

Give an expression that generates all angles coterminal with each angle. Let n represent any...

An ideal Otto cycle has a compression ratio of 10.5, takes in air at 90...

A piece of wire 10 m long is cut into two pieces. One piece is...

Put the following equation of a line into slope intercept form, simplifying all fractions 3x+3y=24

Find the point on the hyperbola xy = 8 that is closest to the point...

Water is pumped from a lower reservoir to a higher reservoir by a pump that...

A piston–cylinder device initially contains 0.07m3 of nitrogen gas at 130 kPa and 180∘. The...

Write an algebraic expression for each word phrase. 4 more than p

A club has 25 members. a) How many ways are there to choose four members...

For each of the sets below, determine whether {2} is an element of that set....

Which expression has both 8 and n as factors?

If repetitions are not permitted (a) how many 3 digit number can be formed from...

To determine the sum of all multiples of 3 between 1 and 1000

On average, there are 3 accidents per month at one intersection. We need to find...

One number is 2 more than 3 times another. Their sum is 22. Find the...

The PMF for a flash drive with X (GB) of memory that was purchased is...

An airplane needs to reach a velocity of 203.0 km/h to takeoff. On a 2000...

A racquetball strikes a wall with a speed of 30 m/s and rebounds with a...

Assuming that the random variable x has a cumulative distribution function,F(x)={0,x<00.25x,0≤x<51,5≤xDetermine the following:a)p(x<2.8)b)p(x>1.5)c)p(x<−z)d)p(x>b)

At t = 0 a grinding wheel has an angular velocity of 24.0 rad/s. It...

How many 3/4's are in 1?

You’re driving down the highway late one night at 20 m/s when a deer steps...

Table salt contains 39.33 g of sodium per 100 g of salt. The U.S. Food...

The constant-pressure heat capacity of a sample of a perfect gas was found to vary...

Coffee is draining from a conical filter into a cylindrical coffepot at the rate of...

Cart is driven by a large propeller or fan, which can accelerate or decelerate the...

A vending machine dispenses coffee into an eight-ounce cup. The amounts of coffee dispensed into...

On an essentially frictionless, horizontal ice rink, a skater moving at 3.0 m/s encounters a...

The gage pressure in a liquid at a depth of 3 m is read to...

Consider a cylindrical specimen of a steel alloy 8.5 mm (0.33 in.) in diameter and...

Calculate the total kinetic energy, in Btu, of an object with a mass of 10...

A 0.500-kg mass on a spring has velocity as a function of time given by...

An Australian emu is running due north in a straight line at a speed of...

Another pitfall cited is expecting to improve the overall performance of a computer by improving...

You throw a glob of putty straight up toward the ceiling, which is 3.60 m...

A 0.150-kg frame, when suspended from a coil spring, stretches the spring 0.070 m. A...

A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips....

A rock climber stands on top of a 50-m-high cliff overhanging a pool of water....

A tank whose bottom is a mirror is filled with water to a depth of...

Two sites are being considered for wind power generation. In the first site, the wind...

0.250 kilogram of water at75.0∘Care contained in a tiny, inert beaker. How much ice, at...

Two boats start together and race across a 60-km-wide lake and back. Boat A goes...

A roller coaster moves 200 ft horizontally and the rises 135 ft at an angle...

A tow truck drags a stalled car along a road. The chain makes an angle...

Consider the curve created by2x2+3y2–4xy=36(a) Show thatdydx=2y−2x3y−2x(b) Calculate the slope of the line perpendicular to...

The current entering the positive terminal of a device is i(t)=6e−2t mA and the voltage...

The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m/s....

Calculate the total potential energy, in Btu, of an object that is 20 ft below...

A chemist in an imaginary universe, where electrons have a different charge than they do...

When jumping, a flea reaches a takeoff speed of 1.0 m/s over a distance of...

Determine the energy required to accelerate a 1300-kg car from 10 to 60 km/h on...

The deepest point in the ocean is 11 km below sea level, deeper than MT....

A golfer imparts a speed of 30.3 m/s to a ball, and it travels the...

Calculate the frequency of each of the following wavelengths of electromagnetic radiation. A) 632.8 nm...

Prove that there is a positive integer that equals the sum of the positive integers...

A hurricane wind blows across a 6.00 m×15.0 m flat roof at a speed of...

If an electron and a proton are expelled at the same time,2.0×10−10mapart (a typical atomic...

The speed of sound in air at 20 C is 344 m/s. (a) What is...

Which of the following functions f has a removable discontinuity at a? If the discontinuity...

A uniform steel bar swings from a pivot at one end with a period of...

A wind farm generator uses a two-bladed propellermounted on a pylon at a height of...

A copper calorimeter can with mass 0.100 kg contains 0.160 kgof water and 0.018 kg...

Jones figures that the total number of thousands of miles that a used auto can...

Assign a binary code in some orderly manner to the 52 playingcards. Use the minimum...

A copper pot with mass 0.500 kg contains 0.170 kg of water ata temperature of...

Ea for a certain biological reaction is 50 kJ/mol, by what factor ( how many...

When a person stands on tiptoe (a strenuous position), the position of the foot is...

A solution was prepared by dissolving 1210 mg of K3Fe(CN)6 (329.2 g/mol) in sufficient waterto...

A 58-kg skier is going down a slope oriented 35 degree abovethe horizontal. The area...

The mechanics at lincoln automotive are reboring a 6-in deepcylinder to fit a new piston....

A 0.48 kg piece of wood floats in water but is found to sinkin alcohol...

A 50-g ice cube at 0oC is heated until 45-g hasbecome water at 100oC and...

A solution containing 6.23 ppm of KMnO4 had a transmittance of 0.195 in a 1.00-cm...

A black body at 7500K consists of an opening of diameter 0.0500mm, looking into an...

A new absolute temperature scale is proposed. On thisscale the ice point of water is...

A 65.0 mm focal length converging lens is 78.0 mm away from a sharp image....

A crate of fruit with mass 35.0 kg and specific heat capacity 3650 J/Kg ....

A freezer has a thermal efficiency of 2.40. Thefreezer is to convert 1.80 kg of...

A horizontal force of 210N is exerted on a 2.0 kg discus as it rotates...

Lead has a specific heat of 0.030 cal/gC. In an insulated container, 300 grams of...

A parachutist relies on air resistance mainly on her parachute to decrease her downward velocity....

The distance between a carbon atom (m=12 u) and an oxygen atom (m + 16...

A car heading north collides at an intersection with a truckheading east. If they lock...

Water stands at a depth H in a large, open tank whose sidewalls are vertical....

The heaviest invertebrate is the giant squid, which is estimated to have a weight of...

Which of the following is a correct comment? */ Comments */ ** Comment ** /*...

The concentrated sulfuric acid we use in the laboratory is 98% H2SO4 by mass. Calculate...

Consider the reaction N2(g)+3H2(g)→2NH3(g) suppose that a particular moment during the reaction molecular hydrogen on...

use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F...

• High School Questions
• College Questions
• Math Solver
• Top Questions 2
• Term of Service
• Payment Policy

Connect with us

Get Plainmath App

• Google Play

E-mail us: [email protected]

Our Service is useful for:

Plainmath is a platform aimed to help users to understand how to solve math problems by providing accumulated knowledge on different topics and accessible examples.

2023 Plainmath. All rights reserved

Category: Practice Questions

Vertical Line Charts Practice Questions

Real Life Linear Graphs Practice Questions

Misleading Graphs Practice Questions

Dual Bar Charts Practice Questions

Composite Bar Charts Practice Questions

Tally Charts Practice Questions

Pictograms Practice Questions

Bar Charts Practice Questions

Splitting the Middle Practice Questions

Invariant Points Practice Questions

Factorising Brackets Practice Questions

Combined Mean Practice Questions

Money Coins Practice Questions

Identities, Formulae, Equations Practice Questions

Cost per Metre Practice Questions

Equating Coefficients Practice Questions

Compass Directions Practice Questions

Profit Practice Questions

Money (Paying) Practice Questions

Quartiles from a List Practice Questions

Money – Midpoint Practice Questions

Money (Estimation) Practice Questions

Reading Tables Practice Questions

Greatest Number (Money) Practice Questions

Midpoint of Two Fractions Practice Questions

Mode from a table practice questions, multipliers practice questions, range from a frequency table practice questions, cost per litre video.

Cost per Litre Practice Questions

Turning Points using Completing the Square Practice Questions

Mean from a Frequency Table Practice Questions

Population Density Practice Questions

Odd and Even Numbers Practice Questions

Fibonacci Practice Questions

Ordering Fractions Practice Questions

VAT Practice Questions

Questionnaires Practice Questions

Reciprocals Practice Questions

Wages Practice Questions

Discounts Practice Questions

Capture Recapture Practice Questions

Regular Payments Practice Questions

Cost per Kg Practice Questions

Income Tax Practice Questions

Gcse revision cards.

5-a-day Workbooks

Primary Study Cards

Privacy Policy

Terms and Conditions

Corbettmaths © 2012 – 2024

You must be logged in to post a comment.

• Math Forum/Help
• Problem Solver
• College Math

Math Practice

Problems for 1st grade, problems for 2nd grade, problems for 3rd grade, problems for 4th grade, problems for 5th grade, problems for 6th grade, problems for 7th grade, problems for 8th grade, problems for 9-12 grade:, quadratic equations.

Views of a function Domain of a function Domain and range Range of a function Inverses of functions Shifting and reflecting functions Positive and negative parts of functions

Line graph intuition Slope of a line Slope intercept form Recognizing slope

Trygonometry

Probabilities, complex numbers.

Mastery-Aligned Maths Tutoring

“The best thing has been the increase in confidence and tutors being there to deal with any misunderstandings straight away."

FREE daily maths challenges

A new KS2 maths challenge every day. Perfect as lesson starters - no prep required!

30 Problem Solving Maths Questions And Answers For GCSE

Sophie Bessemer

Problem solving maths questions can be challenging for GCSE students as there is no ‘one size fits all’ approach. In this article, we’ve compiled tips for problem solving, example questions, solutions and problem solving strategies for GCSE students. 

Since the current GCSE specification began, there have been many maths problem solving exam questions which take elements of different areas of maths and combine them to form new maths problems which haven’t been seen before. 

While learners can be taught to approach simply structured problems by following a process, questions often require students to make sense of lots of new information before they even move on to trying to solve the problem. This is where many learners get stuck.

GCSE MATHS 2025: STAY UP TO DATE Join our email list to stay up to date with the latest news, revision lists and resources for GCSE maths 2025. We’re analysing each paper during the course of the 2025 GCSEs in order to identify the key topic areas to focus on for your revision. GCSE dates 2025 GCSE results 2025 (when published) GCSE results 2024 Analysis of GCSE Maths Paper 1 (2024) Analysis of GCSE Maths Paper 2 (2024) Analysis of GCSE Maths Paper 3 (2024) Summary of ALL GCSE Maths Papers (2024)

How to teach problem solving

In the Ofsted maths review , published in May 2021, Ofsted set out their findings from the research literature regarding the sort of curriculum and teaching that best supports all pupils to make good progress in maths throughout their time in school.

Regarding the teaching of problem solving skills, these were their recommendations:

• Teachers could use a curricular approach that better engineers success in problem-solving by teaching the useful combinations of facts and methods, how to recognise the problem types and the deep structures that these strategies pair to.
• Strategies for problem-solving should be topic specific and can therefore be planned into the sequence of lessons as part of the wider curriculum. Pupils who are already confident with the foundational skills may benefit from a more generalised process involving identifying relationships and weighing up features of the problem to process the information. 
• Worked examples, careful questioning and constructing visual representations can help pupils to convert information embedded in a problem into mathematical notation.
• Open-ended problem solving tasks do not necessarily mean that the activity is the ‘ideal means of acquiring proficiency’. While enjoyable, open ended problem-solving activities may not necessarily lead to improved results.  

If you’re a KS2 teacher needing more support and CPD around teaching reasoning, problem solving & planning for depth we have a whole series of word problems and strategies to teach them available for you.

30 Problem Solving Maths Questions, Solutions & Strategies

Help your students prepare for their math GCSE with these free problem solving maths questions, solutions and strategies.

6 tips to tackling problem solving maths questions

There is no ‘one size fits all’ approach to successfully tackling problem solving maths questions however, here are 6 general tips for students facing a problem solving question:

• Read the whole question, underline important mathematical words, phrases or values.
• Annotate any diagrams, graphs or charts with any missing information that is easy to fill in.
• Think of what a sensible answer may look like. E.g. Will the angle be acute or obtuse? Is £30,000 likely to be the price of a coat?
• Tick off information as you use it.
• Draw extra diagrams if needed.
• Look at the final sentence of the question. Make sure you refer back to that at the end to ensure you have answered the question fully.

There are many online sources of mathematical puzzles and questions that can help learners improve their problem-solving skills. Websites such as NRICH and our blog on SSDD problems have some great examples of KS2, KS3 and KS4 mathematical problems.

Read more: KS2 problem solving and KS3 maths problem solving

In this article, we’ve focussed on GCSE questions and compiled 30 problem solving maths questions and solutions suitable for Foundation and Higher tier students. Additionally, we have provided problem solving strategies to support your students for some questions to encourage critical mathematical thinking . For the full set of questions, solutions and strategies in a printable format, please download our 30 Problem Solving Maths Questions, Solutions & Strategies.

Looking for additional support and resources at KS3? You are welcome to download any of the secondary maths resources from Third Space Learning’s resource library for free. There is a section devoted to GCSE maths revision with plenty of maths worksheets and GCSE maths questions . There are also maths tests for KS3, including a Year 7 maths test , a Year 8 maths test and a Year 9 maths test For children who need more support, our maths intervention programmes for KS3 achieve outstanding results through a personalised one to one tuition approach.

10 problem solving maths questions (Foundation tier)

These first 10 questions and solutions are similar to Foundation questions. For the first three, we’ve provided some additional strategies.

In our downloadable resource, you can find strategies for all 10 Foundation questions .

1) L-shape perimeter 

Here is a shape:

Sarah says, “There is not enough information to find the perimeter.”

Is she correct? What about finding the area?

• Try adding more information – giving some missing sides measurements that are valid. 
• Change these measurements to see if the answer changes.
• Imagine walking around the shape if the edges were paths. Could any of those paths be moved to another position but still give the same total distance?

The perimeter of the shape does not depend on the lengths of the unlabelled edges.

Edge A and edge B can be moved to form a rectangle, meaning the perimeter will be 22 cm. Therefore, Sarah is wrong.

The area, however, will depend on those missing side length measurements, so we would need more information to be able to calculate it.

2) Find the missing point

Here is a coordinate grid with three points plotted. A fourth point is to be plotted to form a parallelogram. Find all possible coordinates of the fourth point.

• What are the properties of a parallelogram?
• Can we count squares to see how we can get from one vertex of the parallelogram to another? Can we use this to find the fourth vertex?

There are 3 possible positions.

3) That rating was a bit mean!

The vertical line graph shows the ratings a product received on an online shopping website. The vertical line for 4 stars is missing.

If the mean rating is 2.65, use the information to complete the vertical line graph.

Strategies 

• Can the information be put into a different format, either a list or a table?
• Would it help to give the missing frequency an algebraic label, x ?
• If we had the data in a frequency table, how would we calculate the mean?
• Is there an equation we could form?

Letting the frequency of 4 star ratings be x , we can form the equation \frac{45+4x}{18+x} =2.65

Giving x=2 

4) Changing angles

The diagram shows two angles around a point. The sum of the two angles around a point is 360°.

Peter says “If we increase the small angle by 10% and decrease the reflex angle by 10%, they will still add to 360°.”

Explain why Peter might be wrong.

Are there two angles where he would be correct?

Peter is wrong, for example, if the two angles are 40° and 320°, increasing 40° by 10% gives 44°, decreasing 320° by 10% gives 288°. These sum to 332°.

10% of the larger angle will be more than 10% of the smaller angle so the sum will only ever be 360° if the two original angles are the same, therefore, 180°.

5) Base and power

The integers 1, 2, 3, 4, 5, 6, 7, 8 and 9 can be used to fill in the boxes. 

How many different solutions can be found so that no digit is used more than once?

There are 8 solutions.

6) Just an average problem 

Place six single digit numbers into the boxes to satisfy the rules.

The mean in maths is 5  \frac{1}{3}

The median is 5

The mode is 3.

How many different solutions are possible?

There are 4 solutions.

2, 3, 3, 7, 8, 9

3, 3, 4, 6, 7, 9

3, 3, 3, 7, 7, 9

3, 3, 3, 7, 8, 8

7) Square and rectangle  

The square has an area of 81 cm 2 . The rectangle has the same perimeter as the square.

Its length and width are in the ratio 2:1.

Find the area of the rectangle.

The sides of the square are 9 cm giving a perimeter of 36 cm. 

We can then either form an equation using a length 2x and width x . 

Or, we could use the fact that the length and width add to half of the perimeter and share 18 in the ratio 2:1. 

The length is 12 cm and the width is 6 cm, giving an area of 72 cm 2 .

8) It’s all prime

The sum of three prime numbers is equal to another prime number.

If the sum is less than 30, how many different solutions are possible?

There are 6 solutions. 

2 can never be used as it would force two more odd primes into the sum to make the total even.

9) Unequal share

Bob and Jane have £10 altogether. Jane has £1.60 more than Bob. Bob spends one third of his money. How much money have Bob and Jane now got in total?

Initially Bob has £4.20 and Jane has £5.80. Bob spends £1.40, meaning the total £10 has been reduced by £1.40, leaving £8.60 after the subtraction.

10) Somewhere between

Fred says, “An easy way to find any fraction which is between two other fractions is to just add the numerators and add the denominators.” Is Fred correct?

Solution 

Fred is correct. His method does work and can be shown algebraically which could be a good problem for higher tier learners to try.

If we use these two fractions \frac{3}{8} and \frac{5}{12} , Fred’s method gives us \frac{8}{20} = \frac{2}{5}

\frac{3}{8} = \frac{45}{120} , \frac{2}{5} = \frac{48}{120} , \frac{5}{12} = \frac{50}{120} . So \frac{3}{8} < \frac{2}{5} < \frac{5}{12}

10 problem solving maths questions (Foundation & Higher tier crossover)

The next 10 questions are crossover questions which could appear on both Foundation and Higher tier exam papers. We have provided solutions for each and, for the first three questions, problem solving strategies to support learners.

11) What’s the difference?

An arithmetic sequence has an nth term in the form an+b .

4 is in the sequence.

16 is in the sequence.

8 is not in the sequence.

-2 is the first term of the sequence.

What are the possible values of a and b ?

• We know that the first number in the sequence is -2 and 4 is in the sequence. Can we try making a sequence to fit? Would using a number line help?
• Try looking at the difference between the numbers we know are in the sequence.

If we try forming a sequence from the information, we get this:

We can now try to fill in the missing numbers, making sure 8 is not in the sequence. Going up by 2 would give us 8, so that won’t work.

The only solutions are 6 n -8 and 3 n -5.

12) Equation of the hypotenuse

The diagram shows a straight line passing through the axes at point P and Q .

Q has coordinate (8, 0). M is the midpoint of PQ and MQ has a length of 5 units.

Find the equation of the line PQ .

• We know MQ is 5 units, what is PQ and OQ ?
• What type of triangle is OPQ ?
• Can we find OP if we know PQ and OQ ?
• A line has an equation in the form y=mx+c . How can we find m ? Do we already know c ?

PQ is 10 units. Using Pythagoras’ Theorem OP = 6

The gradient of the line will be \frac{-6}{8} = -\frac{3}{4} and P gives the intercept as 6.

13) What a waste

Harry wants to cut a sector of radius 30 cm from a piece of paper measuring 30 cm by 20 cm. 

What percentage of the paper will be wasted?

• What information do we need to calculate the area of a sector? Do we have it all?
• Would drawing another line on the diagram help find the angle of the sector?

The angle of the sector can be found using right angle triangle trigonometry.

The angle is 41.81°.

This gives us the area of the sector as 328.37 cm 2 .

The area of the paper is 600 cm 2 .

The area of paper wasted would be 600 – 328.37 = 271.62 cm 2 .

The wasted area is 45.27% of the paper.

14) Tri-polygonometry

The diagram shows part of a regular polygon and a right angled triangle. ABC is a straight line. Find the sum of the interior angles of the polygon.

Finding the angle in the triangle at point B gives 30°. This is the exterior angle of the polygon. Dividing 360° by 30° tells us the polygon has 12 sides. Therefore, the sum of the interior angles is 1800°.

15) That’s a lot of Pi

A block of ready made pastry is a cuboid measuring 3 cm by 10 cm by 15 cm. 

Anne is making 12 pies for a charity event. For each pie, she needs to cut a circle of pastry with a diameter of 18 cm from a sheet of pastry 0.5 cm thick.

How many blocks of pastry will Anne need to buy?

The volume of one block of pastry is 450 cm 3 . 

The volume of one cylinder of pastry is 127.23 cm 3 .

12 pies will require 1526.81 cm 3 .

Dividing the volume needed by 450 gives 3.39(…). 

Rounding this up tells us that 4 pastry blocks will be needed.

16) Is it right?

A triangle has sides of (x+4) cm, (2x+6) cm and (3x-2) cm. Its perimeter is 80 cm.

Show that the triangle is right angled and find its area.

Forming an equation gives 6x+8=80

This gives us x=12 and side lengths of 16 cm, 30 cm and 34 cm.

Using Pythagoras’ Theorem

16 2 +30 2 =1156 

Therefore, the triangle is right angled.

The area of the triangle is (16 x 30) ÷ 2 = 240 cm 2 .

17) Pie chart ratio

The pie chart shows sectors for red, blue and green. 

The ratio of the angles of the red sector to the blue sector is 2:7. 

The ratio of the angles of the red sector to the green sector is 1:3. 

Find the angles of each sector of the pie chart.

Multiplying the ratio of red : green by 2, it can be written as 2:6. 

Now the colour each ratio has in common, red, has equal parts in each ratio.

The ratio of red:blue is 2:7, this means red:blue:green = 2:7:6.

Sharing 360° in this ratio gives red:blue:green = 48°:168°:144°.

18) DIY Simultaneously

Mr Jones buys 5 tins of paint and 4 rolls of decorating tape. The total cost was £167.

The next day he returns 1 unused tin of paint and 1 unused roll of tape. The refund amount is exactly the amount needed to buy a fan heater that has been reduced by 10% in a sale. The fan heater normally costs £37.50.

Find the cost of 1 tin of paint.    

The sale price of the fan heater is £33.75. This gives the simultaneous equations

p+t = 33.75 and 5 p +4 t = 167.

We only need the price of a tin of paint so multiplying the first equation by 4 and then subtracting from the second equation gives p =32. Therefore, 1 tin of paint costs £32. 

19) Triathlon pace

Jodie is competing in a Triathlon. 

A triathlon consists of a 5 km swim, a 40 km cycle and a 10 km run. 

Jodie wants to complete the triathlon in 5 hours. 

She knows she can swim at an average speed of 2.5 km/h and cycle at an average speed of 25 km/h. There are also two transition stages, in between events, which normally take 4 minutes each.

What speed must Jodie average on the final run to finish the triathlon in 5 hours?

Dividing the distances by the average speeds for each section gives times of 2 hours for the swim and 1.6 hours for the cycle, 216 minutes in total. Adding 8 minutes for the transition stages gives 224 minutes. To complete the triathlon in 5 hours, that would be 300 minutes. 300 – 224 = 76 minutes. Jodie needs to complete her 10 km run in 76 minutes, or \frac{19}{15} hours. This gives an average speed of 7.89 km/h.

20) Indices

a 2x × a y =a 3

(a 3 ) x ÷ a 4y =a 32

Find x and y .

Forming the simultaneous equations

Solving these gives

10 problem solving maths questions (Higher tier)

This final set of 10 questions would appear on the Higher tier only. Here we have just provided the solutions. Try asking your learners to discuss their strategies for each question.  

21) Angles in a polygon

The diagram shows part of a regular polygon.

A , B and C are vertices of the polygon. 

The size of the reflex angle ABC is 360° minus the interior angle.

Show that the sum of all of these reflex angles of the polygon will be 720° more than the sum of its interior angles.

Each of the reflex angles is 180 degrees more than the exterior angle: 180 + \frac{360}{n}

The sum of all of these angles is n (180 + \frac{360}{n} ). 

This simplifies to 180 n + 360

The sum of the interior angles is 180( n – 2) = 180 n – 360

The difference is 180 n + 360 – (180 n -360) = 720°

22) Prism and force (Non-calculator)

The diagram shows a prism with an equilateral triangle cross-section.

When the prism is placed so that its triangular face touches the surface, the prism applies a force of 12 Newtons resulting in a pressure of \frac{ \sqrt{3} }{4} N/m^{2}

Given that the prism has a volume of 384 m 3 , find the length of the prism.

Pressure = \frac{Force}{Area}

Area = 12÷ \frac{ \sqrt{3} }{4} = 16\sqrt{3} m 2

Therefore, the length of the prism is 384 ÷ 16\sqrt{3} = 8\sqrt{3} m

23) Geometric sequences (Non-calculator)

A geometric sequence has a third term of 6 and a sixth term of 14 \frac{2}{9}

Find the first term of the sequence.

The third term is ar 2 = 6

The sixth term is ar 5 = \frac{128}{9}

Diving these terms gives r 3 = \frac{64}{27}

Giving r = \frac{4}{3}

Dividing the third term twice by \frac{4}{3} gives the first term a = \frac{27}{8}

24) Printing factory

A printing factory is producing exam papers. When all 10 of its printers are working, it can produce all of the exam papers in 12 days.

For the first two days of printing, 3 of the printers are broken.

At the beginning of the third day it is discovered that 2 more printers have broken down, so the factory continues to print with the reduced amount of printers for 3 days. The broken printers are repaired and now all printers are available to print the remaining exams.

How many days in total does it take the factory to produce all of the exam papers?

If we assume one printer prints 1 exam paper per day, 10 printers would print 120 exam papers in 12 days. Listing the number printed each day for the first 5 days gives:

Day 5: 5 

This is a total of 29 exam papers.

91 exam papers are remaining with 10 printers now able to produce a total of 10 exam papers each day. 10 more days would be required to complete the job.

Therefore, 15 days in total are required.

25) Circles

The diagram shows a circle with equation x^{2}+{y}^{2}=13 .

A tangent touches the circle at point P when x=3 and y is negative.

The tangent intercepts the coordinate axes at A and B .

Find the length AB .

Using the equation  x^{2}+y^{2}=13 to find the y value for P gives y=-2 .

The gradient of the radius at this point is - \frac{2}{3} , giving a tangent gradient of \frac{3}{2} .

Using the point (3,-2) in y = \frac {3}{2} x+c gives the equation of the tangent as y = \frac {3}{2} x – \frac{13}{2}

Substituting x=0 and y=0 gives A and B as (0 , -\frac {13}{2}) and ( \frac{13}{3} , 0)

Using Pythagoras’ Theorem gives the length of AB as ( \frac{ 13\sqrt{13} }{6} ) = 7.812.

26) Circle theorems

The diagram shows a circle with centre O . Points A, B, C and D are on the circumference of the circle. 

EF is a tangent to the circle at A . 

Angle EAD = 46°

Angle FAB = 48°

Angle ADC = 78°

Find the area of ABCD to the nearest integer.

The Alternate Segment Theorem gives angle ACD as 46° and angle ACB as 48°.

Opposite angles in a cyclic quadrilateral summing to 180° gives angle ABC as 102°.

Using the sine rule to find AC will give a length of 5.899. Using the sine rule again to find BC will give a length of 3.016cm.

We can now use the area of a triangle formula to find the area of both triangles.

0.5 × 5 × 5.899 × sin (46) + 0.5 × 3.016 × 5.899 × sin (48) = 17 units 2 (to the nearest integer).

27) Quadratic function

The quadratic function f(x) = -2x^{2} + 8x +11 has a turning point at P .

Find the coordinate of the turning point after the transformation -f(x-3) .

There are two methods that could be used. We could apply the transformation to the function and then complete the square, or, we could complete the square and then apply the transformation.

Here we will do the latter.

This gives a turning point for f(x) as (2,19).

Applying -f(x-3) gives the new turning point as (5,-19).

28) Probability with fruit

A fruit bowl contains only 5 grapes and n strawberries.

A fruit is taken, eaten and then another is selected.

The probability of taking two strawberries is \frac{7}{22} .

Find the probability of taking one of each fruit. 

There are n+5 fruits altogether.

P(Strawberry then strawberry)= \frac{n}{n+5} × \frac{n-1}{n+4} = \frac{7}{22}

This gives the quadratic equation 15n^{2} - 85n - 140 = 0

This can be divided through by 5 to give 3n^{2} - 17n- 28 = 0

This factorises to (n-7)(3n + 4) = 0

n must be positive so n = 7.

The probability of taking one of each fruit is therefore, \frac{5}{12} × \frac{7}{11} + \frac {7}{12} × \frac {5}{11} = \frac {70}{132}

29) Ice cream tub volume

An ice cream tub in the shape of a prism with a trapezium cross-section has the dimensions shown. These measurements are accurate to the nearest cm.

An ice cream scoop has a diameter of 4.5 cm to the nearest millimetre and will be used to scoop out spheres of ice cream from the tub.

Using bounds find a suitable approximation to the number of ice cream scoops that can be removed from a tub that is full.

We need to find the upper and lower bounds of the two volumes. 

Upper bound tub volume = 5665.625 cm 3

Lower bound tub volume = 4729.375 cm 3

Upper bound scoop volume = 49.32 cm 3  

Lower bound scoop volume = 46.14 cm 3  

We can divide the upper bound of the ice cream tub by the lower bound of the scoop to get the maximum possible number of scoops. 

Maximum number of scoops = 122.79

Then divide the lower bound of the ice cream tub by the upper bound of the scoop to get the minimum possible number of scoops.

Minimum number of scoops  = 95.89

These both round to 100 to 1 significant figure, Therefore, 100 scoops is a suitable approximation the the number of scoops.

30) Translating graphs

 The diagram shows the graph of y = a+tan(x-b ).

The graph goes through the points (75, 3) and Q (60, q).

Find exact values of a , b and q .

The asymptote has been translated to the right by 30°. 

Therefore, b=30

So the point (45,1) has been translated to the point (75,3). 

Therefore, a=2

We hope these problem solving maths questions will support your GCSE teaching. To get all the solutions and strategies in a printable form, please download the complete resource .

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost progress.

Since 2013 these personalised one to 1 lessons have helped over 169,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

Related articles

Maths Problem Solving: Engaging Your Students And Strengthening Their Mathematical Skills

Free Year 7 Maths Test With Answers And Mark Scheme: Mixed Topic Questions

What Is A Number Square? Explained For Primary School Teachers, Parents & Pupils

What Is Numicon? Explained For Primary School Teachers, Parents And Pupils

FREE Guide to Maths Mastery

All you need to know to successfully implement a mastery approach to mathematics in your primary school, at whatever stage of your journey.

Ideal for running staff meetings on mastery or sense checking your own approach to mastery.

Privacy Overview

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

Get math help in your language

Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

• the length of the room: L
• the width of the room: W
• the total Area including veranda: A
• the width of the room is half its length: W = ½L
• the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

Question Generator

Enjoy using the site, become a member and enjoy access to all the resources on the site ad-free.

Want Better Math Grades?

✅ Unlimited Solutions

✅ Step-by-Step Answers

✅ Available 24/7

➕ Free Bonuses ($1085 value!)

On this page

• Search IntMath
• Math interactives
• About (site info)
• Uses of Trignometry
• ASCIIMath input, KaTeX output
• ASCIIMath input, LaTeX and KaTeX output
• Send Math in emails
• Syntax for ASCIIMathML
• Math Display Experiments
• Scientific Notebook

Math Problem Solver

Related Sections

Math Tutoring

Need help? Chat with a tutor anytime, 24/7.

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor.

Sign up for free here .

Problem Solver Subjects

Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

• Math Word Problems
• Pre-Algebra
• Geometry Graphing
• Trigonometry
• Precalculus
• Finite Math
• Linear Algebra

Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

Basic Math Solutions

Below are examples of basic math problems that can be solved.

• Long Arithmetic
• Rational Numbers
• Operations with Fractions
• Ratios, Proportions, Percents
• Measurement, Area, and Volume
• Factors, Fractions, and Exponents
• Unit Conversions
• Data Measurement and Statistics
• Points and Line Segments

Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

• Variables, Expressions, and Integers
• Simplifying and Evaluating Expressions
• Solving Equations
• Multi-Step Equations and Inequalities
• Ratios, Proportions, and Percents
• Linear Equations and Inequalities

Algebra Solutions

Below are examples of Algebra math problems that can be solved.

• Algebra Concepts and Expressions
• Points, Lines, and Line Segments
• Simplifying Polynomials
• Factoring Polynomials
• Linear Equations
• Absolute Value Expressions and Equations
• Radical Expressions and Equations
• Systems of Equations
• Quadratic Equations
• Inequalities
• Complex Numbers and Vector Analysis
• Logarithmic Expressions and Equations
• Exponential Expressions and Equations
• Conic Sections
• Vector Spaces
• 3d Coordinate System
• Eigenvalues and Eigenvectors
• Linear Transformations
• Number Sets
• Analytic Geometry

Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

• Algebra Concepts and Expressions Review
• Right Triangle Trigonometry
• Radian Measure and Circular Functions
• Graphing Trigonometric Functions
• Simplifying Trigonometric Expressions
• Verifying Trigonometric Identities
• Solving Trigonometric Equations
• Complex Numbers
• Analytic Geometry in Polar Coordinates
• Exponential and Logarithmic Functions
• Vector Arithmetic

Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

• Operations on Functions
• Rational Expressions and Equations
• Polynomial and Rational Functions
• Analytic Trigonometry
• Sequences and Series
• Analytic Geometry in Rectangular Coordinates
• Limits and an Introduction to Calculus

Calculus Solutions

Below are examples of Calculus math problems that can be solved.

• Evaluating Limits
• Derivatives
• Applications of Differentiation
• Applications of Integration
• Techniques of Integration
• Parametric Equations and Polar Coordinates
• Differential Equations

Statistics Solutions

Below are examples of Statistics problems that can be solved.

• Algebra Review
• Average Descriptive Statistics
• Dispersion Statistics
• Probability
• Probability Distributions
• Frequency Distribution
• Normal Distributions
• t-Distributions
• Hypothesis Testing
• Estimation and Sample Size
• Correlation and Regression

Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

• Polynomials and Expressions
• Equations and Inequalities
• Linear Functions and Points
• Systems of Linear Equations
• Mathematics of Finance
• Statistical Distributions

Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

• Introduction to Matrices
• Linear Independence and Combinations

Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

• Unit Conversion
• Atomic Structure
• Molecules and Compounds
• Chemical Equations and Reactions
• Behavior of Gases
• Solutions and Concentrations

Physics Solutions

Below are examples of Physics math problems that can be solved.

• Static Equilibrium
• Dynamic Equilibrium
• Kinematics Equations
• Electricity
• Thermodymanics

Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

• Step By Step Graphing
• Linear Equations and Functions
• Polar Equations

Looking for the old Mathway Calculator? We've moved it to here .

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

Email Address Sign Up

• Solutions Integral Calculator Derivative Calculator Algebra Calculator Matrix Calculator More...
• Graphing Line Graph Exponential Graph Quadratic Graph Sine Graph More...
• Calculators BMI Calculator Compound Interest Calculator Percentage Calculator Acceleration Calculator More...
• Geometry Pythagorean Theorem Calculator Circle Area Calculator Isosceles Triangle Calculator Triangles Calculator More...
• Tools Notebook Groups Cheat Sheets Worksheets Study Guides Practice Verify Solution

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Prove That Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Coterminal Angle Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Number Line

• x^{2}-x-6=0
• -x+3\gt 2x+1
• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• Practice Makes Perfect Learning math takes practice, lots of practice. Just like running, it takes practice and dedication. If you want...

We want your feedback

Please add a message.

Message received. Thanks for the feedback.

Open Middle®

Math Practice

Geogebra math practice.

Math Practice is a tool for mastering algebraic notation. It supports students in their step-by-step math work, let's them explore different solution paths, and helps build confidence, fluency, and understanding.

Getting started as a teacher or student

Enhance your skills

Immerse yourself in the world of algebraic problems to fine-tune your mathematical abilities and elevate your skillset

Linear equations

Order of Operations

Algebraic Expressions

Polynomials

Unlocking the key elements

Use interactive notation to build comfort and fluency with algebraic transformations

Get adaptive hints and in-the-moment feedback to explore different solution paths

Get comprehensive help and guidance focused on improving understanding of basic concepts

Explore features

If you require any guidance on how to use GeoGebra Math Practice, explore the articles in our help center.

Guided tutorials

If you are new to using this tool, we offer easy-to-follow guided tutorials.

6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

Your content has been saved!

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

You are using an outdated browser. Upgrade your browser today or install Google Chrome Frame to better experience this site.

Opening up students’ mathematical thinking using open middle math problems

Back when we were in schools every day, we were constantly on the lookout for resources to support rich, rigorous, mathematical learning. Although many of the teachers we worked to support liked the concrete-representational-abstract framework of their math curriculum, we frequently heard them express frustration over the rote practice problems provided in most of their resources. Teachers were spending a lot of time looking for content that went beyond a standard worksheet. They were telling us they wanted activities that would spark student excitement and encourage deeper thinking, like open middle math problems.

When paired with number talks , open middle problems can help teachers embed rich conversation throughout math class.

Defining the problem

So, what exactly are open middle math problems? In an open middle problem, students are given starting information and either an answer or a definition of the answer. By “definition of the answer,” we mean that students are given a description of the type of answer toward which they are driving. For example, many open middle math problems ask students to find the least or greatest solution or a number that is closest to a given number. How they get from start to end is up to them. Rather than following a solution path that was stepped out for them by the teacher, the path is completely under their control.

The best way to understand an open middle math problem may be to compare one to a typical textbook problem for the same topic. For a lesson on three-digit subtraction, a standard textbook might ask students to find 539 – 286 either as a straight computation item or embedded within a word problem. By contrast, an open middle math problem for the same topic asks students to use the digits 1–9, at most one time each, to make two three-digit numbers with a difference as close to 329 as possible.

Although understanding of place value, regrouping, the relative magnitude of numbers, and the relationship between addition and subtraction underlie both types of problems, with the more traditional problem, a student can answer correctly by following a set of learned steps without truly understanding the conceptual underpinnings. However, students must actively engage with these concepts to solve the open middle math problem. The open structure forces students to consider these concepts and constantly make decisions and adjust their solution path. Here are some of the questions students might consider when solving the open middle math problem:

• What strategies can be applied to solve? Should I subtract? Should I add up? Can estimation help?
• Should I focus on a certain place value for each number first, or should I create one number completely?
• Is it possible to make two numbers where regrouping will not be required?
• Is it possible to get exactly 329?

Open middle problems can also be word problems . While some open middle math problems are structured this way, others can easily be turned into word problems. For example, the three-digit subtraction problem above can be rewritten as a word problem such as this: Jared and Suni are playing a video game. Each of their scores were three-digit numbers. Jared’s score was 329 less than Suni’s. Use the digits 1–9, at most one time each, to make two three-digit numbers that could be Jared and Suni’s scores.

It’s the journey, not the destination

Because of the way they are designed, open middle math problems emphasize process over product. While students are still expected to arrive at a defensible answer, open middle problems push students to utilize a variety of mathematical skills and understandings to wrestle with the problem. The goal is not to be the quickest to follow a predetermined solution path but, rather, to allow for a more iterative approach where students continually test and revise strategies. Dan Meyer terms this “patient problem solving,” where students bring all their mathematical knowledge and intuition to bear on solving non-routine problems. The active engagement that this engenders supports rich mathematical thinking and discussion amongst students.

One of the co-developers of open middle problems, Nanette Johnson, describes their value this way: “This structure requires students to prove to themselves and others that they have truly found the best possible answer. Rather than putting down their pencils and saying, ‘I’m done,’ students continue to think, argue, and work. They develop the habit of making multiple attempts to solve the problem, each time wondering if they can come up with an even better solution than their last.”

The fact that students can arrive at different answers naturally prompts discussion between students of how they approached the problem, why they think their answer best meets the criteria, and whether other answers could get closer to the goal. As Johnson states, this serves to foster increased engagement and persistence. It also provides a vehicle for the rich mathematical discourse we were seeking in our districts.

Low floors, high ceilings, and big concepts

By design, open middle math problems allow easy entry for all students, but they also provide the flexibility to allow students to stretch themselves. For this reason, open middle problems have been described as easy in terms of getting an answer, but not so easy in terms of getting the ideal or “best” answer. Take, for example, that problem that challenges students to use the digits 1–9, at most one time each, to create two mixed numbers with the least possible difference. Students are given a template like the one on the Open Middle website in which to write their numbers.

Students who are still developing their understanding of fraction operations might take a simple approach of having the digits in the second mixed number be one less than the digits in the first mixed number: 9 5/7 – 8 4/6. This results in a difference of 1 1/21. Another student might opt to use the greatest digits available, 8 and 9, to make the denominators of the fractions, knowing that the greater number of parts a fraction is divided into, the smaller it is. Perhaps they create the expression 2 7/9 – 1 6/8 , which results in a difference of 1 1/36. At this point, the teacher might ask these two students which of their answers is the least, sparking a conversation about the relative magnitude of the numbers, benchmark fractions, and estimation. The teacher could then push students by challenging them to see if it is possible to get an answer that is less than one.

A student further along in their understanding of fractions might recognize that the need to regroup could reduce the size of the final number. Perhaps they create the problem 3 1/9 – 2 7/8 with the result of 17/72. Student interactions or well-placed teacher questions, such as, “Is it possible to get an answer of 0?” might prompt some students to realize using improper fractions can further reduce the magnitude of the answer, indeed all the way down to zero.

The nature of the problem allows all students entry without limiting students with more advanced knowledge of the involved concepts. Rich conversations and strategic teacher questions help all students expand and grow their understanding of concepts beyond the idea of fraction subtraction, while providing teachers insights into each students’ level of understanding of these concepts. In the iterative process of testing ideas, discussing different approaches, and trying new solution paths, students explore far deeper concepts of the meaning of fractions, number magnitude, and equivalent forms.

Impact of using open middle math problems

While there is no research that specifically examines open middle problems, there are studies that show the efficacy of components of the approach. In a research paper on equitable teaching approaches to math , Jo Boaler and Megan Staples call out that several studies have demonstrated that “conceptually oriented mathematics materials, taught well and with consistency, have shown higher and more equitable results for participating students than procedure-oriented curricula taught using a demonstration and practice approach.” Open middle math problems support such a conceptual orientation and, by design, break the “demonstration and practice” model.

Having students discuss and compare different solution approaches is a critical component of open math problems. Although students are not working in groups, they are also not working in isolation. Part of the iterative process of using open middle math problems includes time to share and discuss the relative merits of various solution approaches and use these to iterate new approaches. Actively comparing solution methods has been shown to increase procedural understanding , particularly for lower achieving students. Another study found that students who learned by comparing alternative solution approaches demonstrated greater conceptual knowledge and more flexible problem-solving than students who reflected on different approaches one at a time.

Open middle math problems are well aligned to NCTM’s process standards . They provide rich opportunities to practice:

• Building new mathematical knowledge through problem-solving
• Monitoring and reflecting on the problem-solving process
• Making and exploring mathematical conjectures
• Developing, evaluating, and communicating mathematical reasoning and analyzing the reasoning of others

Getting started

When working on incorporating using open middle math problems in your classroom, we encourage you to determine how to integrate them into existing math lesson plans so they aren’t one more thing you feel you have to do. Consider, for example, how open middle math tasks build off number talks but take you and your students to the next level.

Both open middle math problems and number talks encourage rich dialogue between students. While number talks are conversation-base only, open middle math problems involve both conversation and written responses. Number talks are brief 10–15-minute warmups, while open middle math problems give students more time to explore their thinking and the thinking of others. They also provide teachers with a written record of this thinking and how it evolves over the course of iterative problem-solving.

You might also try working on open middle problems yourself, which can help you understand the power of the approach.

Adding doesn’t necessarily mean more

We also encourage you to focus on potential roadblocks to making open middle math problems a part of your classroom. In our experience, time can be a big concern. If that’s worrying you as well, we suggest you focus on the relative merit of having students deeply engage in a single rich mathematical problem in place of completing 10 rote problems. This can help you move away from a “mile-wide, inch deep” approach to math.

Remember, too, that these problems do not need to be completed in a day. Take the time to let students work through possible solutions, share, discuss, and revise over the course of one or more days.

Finally, time you previously spent looking for better resources could be repurposed into unpacking open middle math problems to prepare you for facilitating a rich classroom discussion. Here are some questions to get you started:

• What big mathematical ideas do you hope students will bring into discussions?
• What models or mathematical terms might students use in their explanations?
• What are examples of clear justification or correct reasoning you hope to see?
• What are some possible solution approaches and what do each of them reveal about students’ understanding of the underlying concepts?

Give it a try!

Adding open middle tasks to your math routines can truly deepen both math conversations and student engagement in your classroom. Ready to try them yourself? Use these links to get more information about open middle math problems and supporting mathematical writing and discourse.

Open middle resources

• Open Middle This site, created by Open Middle co-creators Nanette Johnson and Robert Kaplinsky, contains a wealth of open middle tasks organized by grade and domain as well as worksheets in English, French, and Spanish and printable number tiles.
• Open Middle Exercises GeoGebra has converted many of the paper versions of open middle exercises into interactive versions.
• Open Middle Math: Problems that Unlock Student Thinking, Grades 6–12 You can preview Robert Kaplinsky’s book about open middle math here.

Classroom discussion and mathematical writing resources

• “4 ways to engage students with writing in math class” This article delves into a different way to support and encourage student writing about math, including how to write high-quality explanations for how they solved a problem.
• “Eliciting, supporting, and guiding the math: Three key functions of the teacher’s role in facilitating meaningful mathematical discourse” This piece defines what makes math discourse meaningful, why it’s important, and what the teacher’s role is in facilitating such discourse.
• “Mathematical discourse in the classroom: A prime time for discussion” This article provides strategies for facilitating meaningful mathematical discussion with the goal of helping students develop a deeper understanding of mathematical concepts.
• “Using the 5 practices in mathematics teaching” This article provides an overview of how teachers can successfully orchestrate classroom discussions by anticipating possible student solutions, monitoring their work, selecting and sequencing how students share their work, and asking questions that help students make connections between mathematical ideas.

Recommended for you

Math enrichment for all: 3 ways to engage all learners in deep mathematical thinking

6 tips for supporting problem-based learning in your math classroom

7 tips for encouraging student discourse about math with number talks

• View all posts

Reading differentiation made easy

MAP Reading Fluency now includes Coach, a virtual tutor designed to help students strengthen reading skills in as little as 30 minutes a week.

Helping students grow

Students continue to rebound from pandemic school closures. NWEA® and Learning Heroes experts talk about how best to support them here on our blog, Teach. Learn. Grow.

See the post

Put the science of reading into action

The science of reading is not a buzzword. It’s the converging evidence of what matters and what works in literacy instruction. We can help you make it part of your practice.

Get the guide

Support teachers with PL

High-quality professional learning can help teachers feel invested—and supported—in their work.

Read the article

Content disclaimer:

Teach. Learn. Grow. includes diverse perspectives that are meant to be a resource to educators and leaders across the country and around the world. The views expressed are those of the authors and do not necessarily represent those of NWEA.

September 12, 2024

Introducing OpenAI o1-preview

A new series of reasoning models for solving hard problems. Available now.

Update on September 17, 2024: Rate limits are now 50 queries per week for o1-preview and 50 queries per day for o1-mini.

We've developed a new series of AI models designed to spend more time thinking before they respond. They can reason through complex tasks and solve harder problems than previous models in science, coding, and math. Today, we are releasing the first of this series in ChatGPT and our API. This is a preview and we expect regular updates and improvements. Alongside this release, we’re also including evaluations for the next update, currently in development.

How it works

We trained these models to spend more time thinking through problems before they respond, much like a person would. Through training, they learn to refine their thinking process, try different strategies, and recognize their mistakes. 

In our tests, the next model update performs similarly to PhD students on challenging benchmark tasks in physics, chemistry, and biology. We also found that it excels in math and coding. In a qualifying exam for the International Mathematics Olympiad (IMO), GPT-4o correctly solved only 13% of problems, while the reasoning model scored 83%. Their coding abilities were evaluated in contests and reached the 89th percentile in Codeforces competitions. You can read more about this in our technical research post .

As an early model, it doesn't yet have many of the features that make ChatGPT useful, like browsing the web for information and uploading files and images. For many common cases GPT-4o will be more capable in the near term.

But for complex reasoning tasks this is a significant advancement and represents a new level of AI capability. Given this, we are resetting the counter back to 1 and naming this series OpenAI o1.

As part of developing these new models, we have come up with a new safety training approach that harnesses their reasoning capabilities to make them adhere to safety and alignment guidelines. By being able to reason about our safety rules in context, it can apply them more effectively. 

One way we measure safety is by testing how well our model continues to follow its safety rules if a user tries to bypass them (known as "jailbreaking"). On one of our hardest jailbreaking tests, GPT-4o scored 22 (on a scale of 0-100) while our o1-preview model scored 84. You can read more about this in the system card and our research post .

To match the new capabilities of these models, we’ve bolstered our safety work, internal governance, and federal government collaboration. This includes rigorous testing and evaluations using our Preparedness Framework (opens in a new window) , best-in-class red teaming, and board-level review processes, including by our Safety & Security Committee. To advance our commitment to AI safety, we recently formalized agreements with the U.S. and U.K. AI Safety Institutes. We've begun operationalizing these agreements, including granting the institutes early access to a research version of this model. This was an important first step in our partnership, helping to establish a process for research, evaluation, and testing of future models prior to and following their public release.

Whom it’s for

These enhanced reasoning capabilities may be particularly useful if you’re tackling complex problems in science, coding, math, and similar fields. For example, o1 can be used by healthcare researchers to annotate cell sequencing data, by physicists to generate complicated mathematical formulas needed for quantum optics, and by developers in all fields to build and execute multi-step workflows. 

OpenAI o1-mini

The o1 series excels at accurately generating and debugging complex code. To offer a more efficient solution for developers, we’re also releasing OpenAI o1-mini , a faster, cheaper reasoning model that is particularly effective at coding. As a smaller model, o1-mini is 80% cheaper than o1-preview, making it a powerful, cost-effective model for applications that require reasoning but not broad world knowledge. 

How to use OpenAI o1

ChatGPT Plus and Team users will be able to access o1 models in ChatGPT starting today. Both o1-preview and o1-mini can be selected manually in the model picker, and at launch, weekly rate limits will be 30 messages for o1-preview and 50 for o1-mini. We are working to increase those rates and enable ChatGPT to automatically choose the right model for a given prompt.

ChatGPT Enterprise and Edu users will get access to both models beginning next week.  Developers who qualify for API usage tier 5 (opens in a new window) can start prototyping with both models in the API today with a rate limit of 20 RPM. We’re working to increase these limits after additional testing. The API for these models currently doesn't include function calling, streaming, support for system messages, and other features. To get started, check out the API documentation (opens in a new window) .

We also are planning to bring o1-mini access to all ChatGPT Free users . 

What’s next

This is an early preview of these reasoning models in ChatGPT and the API. In addition to model updates, we expect to add browsing, file and image uploading, and other features to make them more useful to everyone. 

We also plan to continue developing and releasing models in our GPT series, in addition to the new OpenAI o1 series. 

• Try it in ChatGPT Plus (opens in a new window)
• Try it in the API (opens in a new window)