problem solving involving real numbers pdf

1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

• ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
• ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• ⓐ 7 = 7 1 7 = 7 1
• ⓑ 0 = 0 1 0 = 0 1
• ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• ⓐ − 5 7 − 5 7
• ⓑ 15 5 15 5
• ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

• ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
• ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
• ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
• ⓐ 68 17 68 17
• ⓑ 8 13 8 13
• ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• ⓑ 33 9 33 9
• ⓓ 17 34 17 34
• ⓔ 0.3033033303333 … 0.3033033303333 …
• ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

• ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

• ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• ⓐ 7 77 7 77
• ⓒ 4.27027002700027 … 4.27027002700027 …
• ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• ⓐ − 10 3 − 10 3
• ⓒ − 289 − 289
• ⓓ −6 π −6 π
• ⓔ 0.615384615384 … 0.615384615384 …
• ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
• ⓑ 5 5 is positive and irrational. It lies to the right of 0.
• ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
• ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
• ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
• ⓑ −11.411411411 … −11.411411411 …
• ⓒ 47 19 47 19
• ⓓ − 5 2 − 5 2
• ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• ⓔ 3.2121121112 … 3.2121121112 …

• ⓐ − 35 7 − 35 7
• ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Step 1. Simplify any expressions within grouping symbols.
• Step 2. Simplify any expressions containing exponents or radicals.
• Step 3. Perform any multiplication and division in order, from left to right.
• Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
• ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
• ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
• ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
• ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

• ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
• ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
• ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
• ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
• ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
• ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
• ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
• ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
• ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
• ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
• ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
• ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
• ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
• ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
• ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
• ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
• ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
• ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
• ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• ⓑ 4 3 π r 3 4 3 π r 3
• ⓒ 2 m 3 n 2 2 m 3 n 2

• ⓐ 2 π r ( r + h ) 2 π r ( r + h )
• ⓑ 2( L + W )
• ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

• ⓐ x = 0 x = 0
• ⓑ x = 1 x = 1
• ⓒ x = 1 2 x = 1 2
• ⓓ x = −4 x = −4
• ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
• ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
• ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
• ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

• ⓐ y = 2 y = 2
• ⓑ y = 0 y = 0
• ⓒ y = 2 3 y = 2 3
• ⓓ y = −5 y = −5

Evaluate each expression for the given values.

• ⓐ x + 5 x + 5 for x = −5 x = −5
• ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
• ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
• ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
• ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
• ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
• ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
• ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
• ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
• ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
• ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
• ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
• ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
• ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
• ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
• ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
• ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
• ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
• ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
• ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
• ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
• ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
• ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
• ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
• ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

• Simplify an Expression.
• Evaluate an Expression 1.
• Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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• Publication date: Dec 21, 2021
• Location: Houston, Texas
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• Section URL: https://openstax.org/books/college-algebra-2e/pages/1-1-real-numbers-algebra-essentials

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Worksheet on Word Problems on Rational Numbers

Practice the questions given in the worksheet on word problems on rational numbers. The questions are related to various types of word problems on four fundamental operations on rational numbers.

1.  From a rope 11 m long, two pieces of lengths 13/5 m and 33/10 m are cut off. What is the length of the remaining rope?  2.  A drum full of rice weighs 241/6 kg. If the empty drum weighs 55/4 kg, find the weight of rice in the drum.  3.  A basket contains three types of fruits weighing 58/3 kg in all. If 73/9 kg of these be apples, 19/6 kg be oranges and the rest pears. What is the weight of the pears in the basket?  4.  On one day a rickshaw puller earned $80. Out of his earnings he spent $68/5 on tea and snacks, $51/2 on food and $22/5 on repairs of the rickshaw. How much did he save on that day? 

5. Find the cost of 17/5 meters of cloth at $147/4 per meter. 6. A car is moving at an average speed of 202/5 km/hr. How much distance will it cover in 15/2 hours? 7. Find the area of a rectangular park which is 183/5 m long and 50/3 m broad. 8. Find the area of a square plot of land whose each side measures 17/2 meters. 9. One liter of petrol costs $187/4. What is the cost of 35 liters of petrol? 10. An airplane covers 1020 km in an hour. How much distance will it cover in 25/6 hours? 11. The cost of 7/2 meters of cloth is $231/4. What is the cost of one meter of cloth? 12. A cord of length 143/2 m has been cut into 26 pieces of equal length. What is the length of each piece? 13. The area of a room is 261/4 m \(^{2}\) . If its breadth is 87/16 meters, what is its length? 14. The product of two rational numbers is 48/5. If one of the rational number is 66/7, find the other rational number. 15. Rita had $300. She spent 1/3 of her money on notebooks and 1/4 of the remainder on stationery items. How much money is left with her? 16. Adrian earns $16000 per month. He spends 1/4 of his income on food; 3/10 of the remainder on house rent and 5/21 of the remainder on the education of children. How much money is still left with him?

Answers for the worksheet on word problems on rational numbers are given below to check the exact answers of the above rational problems.

2. 317/12 kg

3. 145/18 kg

5. $2499/20

7. 610 m \(^{2}\)

8. 289/4 m \(^{2}\)

10. 4250 km

● Rational Numbers - Worksheets

Worksheet on Rational Numbers

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Operations on Rational Expressions

Objective Questions on Rational Numbers

Math Homework Sheets 8th Grade Math Practice From Worksheet on Word Problems on Rational Numbers to HOME PAGE

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● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

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• Preparatory

Lesson Plan: Properties of Operations over the Real Numbers Mathematics • Second Year of Preparatory School

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• Remaining Seats: 7

This lesson plan includes the objectives, prerequisites, and exclusions of the lesson teaching students how to solve problems involving operations and properties of operations on real numbers.

Students will be able to

• closure ( 𝑎 + 𝑏 ∈ ℝ and 𝑎 𝑏 ∈ ℝ ),
• commutativity ( 𝑎 + 𝑏 = 𝑏 + 𝑎 and 𝑎 𝑏 = 𝑏 𝑎 ),
• associativity ( ( 𝑎 + 𝑏 ) + 𝑐 = 𝑎 + ( 𝑏 + 𝑐 ) and ( 𝑎 ⋅ 𝑏 ) ⋅ 𝑐 = 𝑎 ⋅ ( 𝑏 ⋅ 𝑐 ) ),
• identity (or neutral) elements ( 𝑎 + 0 = 𝑎 and 𝑎 ⋅ 1 = 𝑎 ),
• inverse elements ( 𝑎 − 𝑎 = 0 and 𝑎 ⋅ 1 𝑎 = 1 for 𝑎 ≠ 0 ),
• understand that subtraction and division are neither associative nor commutative.

Prerequisites

Students should already be familiar with

• the set of real numbers.

Students will not cover

• simplifying the addition, subtraction, multiplication, and division of radical expressions (including rationalizing denominators).

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Real numbers - practice problems

Number of problems found: 162.

• all math problems 19044
• numbers 6340
• real numbers 162
• triangle 33
• square root 32
• right triangle 30
• equation 25
• Pythagorean theorem 23
• fractions 20
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• area of a shape 17

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7.1.2B Problem Solving with Rational Numbers

Standard 7.1.2.

Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest.

Use proportional reasoning to solve problems involving ratios in various contexts.

For example : A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this is how recipes are often given in large institutions, such as hospitals). How much flour and milk would be needed with 1 cup of sugar?

Standard 7.1.2 Essential Understandings

In this standard, students will develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying and dividing with negative numbers.

The focus of instruction at the 7th grade level is on being able to comfortably translate between decimal and fractional forms of a number for both positive and negative values. Students should be able to compare numbers and manipulate the values to derive other forms of the numbers to make comparing less inhibiting and more accessible. Students will also use ratios and proportional reasoning to solve problems in various contexts. Students will be able to use information given to help find missing values. Their knowledge of equivalent fractions and scaling will enable them to use ratios and solve proportions.

All Standard Benchmarks

7.1.2.1 Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents. 7.1.2.2 Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. 7.1.2.3 Understand that calculators and other computing technologies often truncate or round numbers. 7.1.2.4 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. 7.1.2.5 Use proportional reasoning to solve problems involving ratios in various contexts. 7.1.2.6 Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.

7.1.2 Group B - Problem Solving with Rational Numbers

7.1.2.4 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. 7.1.2.5 Use proportional reasoning to solve problems involving ratios in various contexts. For example, a recipe calls for milk, flour and sugar in a ratio of 4:6:3. (This is how recipes are often given in large institutions, such as hospitals.) How much flour and milk would be needed with 1 cup of sugar?

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Understand that simple interest does not use interest earned as new principal, but compound interest does;
• Model addition and subtraction of integers with physical materials and the number line;
• Perform calculations with and without the use of a calculator;
• Understand the rules for calculating with positive and negatives;
• Understand calculating with positive exponents;
• Scale values up and down;
• See the relationship as a proportional relationship;
• Use ratios accurately;
• Compare ratios;
• Be able to differentiate mathematical characteristics of proportional thinking from nonproportional contexts;
• Know the mathematical characteristics of proportional situations.

Work from previous grades that supports this new learning includes:

• Use and read output on a calculator;
• Know how to change percents to decimals and decimals to percents;
• Know how to calculate a percent of a number, such as 25% of 1000;
• Use and find percents;
• Use and find fractions and equivalent values;
• Multiply and divide;
• Input into a calculator, using correct keystrokes;
• Perform mental math;
• Understand equivalent fractions;
• Scale up and down;
• Use ratios;
• Know multiplication facts to 12's;
• Be proficient at problem solving ;
• Know how to work backwards;
• Understand and use the terms including numerator, denominator, greatest common factor and least common multiple;
• Know how to simplify fractions.
• Make use of estimation strategies.

NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems:

• Work flexibly with fractions, decimals, and percents to solve problems;
• Understand and use ratios and proportions to represent quantitative relationships.

Understand meanings of operations and how they relate to one another:

• Understand the meaning and effects of arithmetic operations with fractions decimals, and integers;
• Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

Compute fluently and make reasonable estimates:

• Select appropriate methods and tools for computing for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situations and apply the selected methods;
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

Common Core State Standards (CCSS)

7.NS ( The Number System ) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

• 7.NS. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram;
• 7.NS.1.b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts;
• 7.NS.1.c . Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts;
• 7.NS.1.d. Apply properties of operations as strategies to add and subtract rational numbers;
• 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers;
• 7.NS.2a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts;
• 7.NS.2.c. Apply properties of operations as strategies to multiply and divide rational numbers;
• 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.

6.NS (The Number System) Apply and extend previous understandings of numbers to the system of rational numbers.

• 6.NS.7c. Understand ordering and absolute value of rational numbers. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
• 6.EE (Expressions and Equations) Apply and extend previous understandings of arithmetic to algebraic expressions.

6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.

7.EE (Expressions and Equations) Use properties of operations to generate equivalent expressions.

• 7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
• 7.RP (Ratios and Proportional Relationships) Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Misconceptions

Student misconceptions and common errors.

• Students may not realize that finding the percent of a number always involves division.
• Students incorrectly enter the percent into the calculator instead of the decimal value, and thus get a value that does not make sense. Many students do not know how to use the % key on a calculator.
• Students get confused with the difference between simple and compound interest.
• Students may not realize that using a proportion is the only way to solve these problems.
• Students want to use addition to get equivalent values, not multiplication.
• Students forget how to multiply with fractional values.
• If using a fractional scale factor (cutting a recipe in $\frac{1}{2}$), students may just add $\frac{1}{2}$ to the values they have, instead of dividing all the values by 2.

In the Classroom

In this vignette, students will mix juice.

Mixing Juice

Teacher: How many of you have made orange juice from a can before?

Student 1: I make it all the time.

Student 2: My mom makes ours.

Others raise their hands to indicate they have either made it or helped make it.

Teacher: When you made juice or helped, what did you have to do?

Student 3: Make sure we didn't add too much water or it wouldn't taste very good.

Teacher: Anything else?

Student 2: Make sure we add enough water or it would be too sour and strong tasting.

Teacher: Today we are going to look at some different recipes for different juice mixes. Each recipe is different from the others, so we are going to use our math skills and see if we can decide which juice would be the most orangey, and which would be the least orangey.

Student 1: Won't it depend on the brand of juice we use, too?

Teacher: You are right; that could be a factor, and that could be something we explore at a different time, but for today, we are going to assume that all the juices are the same brand, so that won't be a factor.

Teacher: OK, here is your information. Cam and Scott are in charge of making juice for the 200 students at camp. They mix water and frozen concentrate to make the juice. They were given the following recipes from different campers:

Teacher: OK, your task is to figure out which one is the most orangey. Good luck!

As the teacher circulates around, she hears some of the following conversations.

Student 1 : I think Mix V will be the most orangey because it has 5 cups of concentrate, and no other juice has that much, so it has to be the most orangey.

Student 2 : I think Mix S and Mix T will be the same because they each have only one more cup of concentrate than juice, so they should taste the same, shouldn't they?

Teacher: Well, how about if we use some concepts we've talk about in class before. What could we do that might help us?

Student 3: Write them as fractions?

Student 2: Find out what percent of each is concentrate?

Student 1: We could write ratios for each recipe.

Teacher: OK. It sounds like we have many ideas. There are multiple ways to do this problem. Just make sure you understand what you are doing, and if you get an answer, make sure you know what it represents in terms of the problem.

Student 1: Well for Mix S, $\frac{2}{3}$ of the juice is concentrate.

Teacher: Let's look at that again. While fractions offer one way of working on this problem, what you said is not accurate. Remember what you know about fractions. If you say that $\frac{2}{3}$ of the juice is concentrate, then which number in the fraction tells you how many total cups are in the juice?

Student 2: Well, the denominator is the total, so the total would be 3 cups. In Mix S, though, there are 3 cups of concentrate AND 2 cups of water, so there's really 5 cups of ingredients in the juice.

Teacher: So is $\frac{2}{3}$ of Mix S concentrate?

Student 1: No, I guess it isn't. But shouldn't we be able to use the fraction $\frac{2}{3}$?

Teacher: Let's talk about that. If we use $\frac{2}{3}$, what does the 2 represent?

Student 3: The 2 cups of water.

Teacher: And what does the 3 represent?

Student 4: The 3 cups of concentrate.

Teacher: So if we wanted to know what fraction is concentrate, what fraction would we use?

Student 2:  $\frac{2}{5}$, because 5 is the total amount of cups in the juice and 2 is the number of cups of concentrate.

Teacher: OK. Keep working and I'll come back to you. Again, we are trying to figure out which one is the most orangey.

The teacher continues to circulate around the room, listening to conversations that different groups are having.

Teacher: Who has an idea here? Which mix does your group think is the most orangey?

Student 2: We think it is Mix T. We set up ratios of concentrate to water, and compared the decimal values. So this is what we got:

Teacher: What do those decimal values represent?

Student 2: The ratio of concentrate to water.

Teacher: Did anyone do this another way?

Student 3: We found out what percent of the juice was concentrate. So here is what we got. We agree that Mix T is the most orangey.

Teacher: So, we can see from these two strategies that Mix T is the most orangey. Which one is the least orangey?

Student 1: Mix U. If you look at the percent that is concentrate, we can see that Mix U has the least percent of concentrate, so it is the least orangey.

Teacher: Let's move on to the next part of the problem. Assume each camper gets $\frac{1}{2}$ cup of juice. For each mix, you need to now figure out how many batches are needed to be made to serve all of the campers. Remember, there are 200 campers.

Student 4: I think we need to first figure out how many cups are in each batch of juice.

Teacher: Why would you need to know that?

Student 4: So we can scale the recipe up to make enough to feed the campers.

Teacher: Good thought process. Go ahead and give it a try and let me know how it goes.

Student 3: Can't we just multiply 200 by $\frac{1}{2}$ to get 100? We need 100 cups of juice made.

Teacher: OK. You are right that you need 100 cups, but that is 100 cups of juice. That doesn't say how many cups of each of the 2 ingredients you need.

Student 3: Oh. That's right. We need to do that. Hmm. Maybe putting the information into tables could help us.

Teacher: Nice. Any other way this could have been figured out?

Student 4: We used proportional reasoning to solve it. We knew that we needed 100 cups total. We also knew how many batches we needed. So, we knew we needed 100 cups total, and there were 5 total cups in the batch made. Since we needed two cups of concentrate, then we can scale that up. Here's the proportion we set up:

         $\frac{2}{5}$ = $\frac{x}{100}$

Student 4: So, we can solve and get 40 for x, just like the other group did. So, we need 40 cups of concentrate and 60 cups of water. We did that same process for the rest of them, too, like our table shows. By doing it our way, we won't have as much left over, because they rounded the number of batches, and we set up proportions so we have a closer number of cups that would be needed. We still have Mix T using the most concentrate in proportion to the amount of water used.

Teacher: OK. Nice work! To finish the lesson, here is your final task: Which of the following will taste most orangey: 2 cups of concentrate and 3 cups of water; 4 cups of concentrate and 6 cups of water; or 10 cups of concentrate and 15 cups of water?

The students use various strategies that were discussed in class.

Teacher: Anyone have an answer?

Student 2 : I think they are all the same orangey-ness?

Teacher: How could that be if they all have different numbers of concentrate and water?

Student 1: Because when we compared all of the ratios, they were equivalent: 2 to 3 is equivalent to 4 to 6, is equivalent to 10 to 15. If we write them all as fractions we can see that even better.

Teacher: Can you come up to the board and show us what you are doing?

Student 1: Sure. Here is what I did:

$\frac{2}{3}$ = $\frac{4}{6}$ = $\frac{10}{15}$

They all simplify to being the same concentrate, $\frac{2}{3}$, so, they are all proportional to one another.

Teacher: Nice job today. We will review this tomorrow.

Teacher Notes

• To find percent increase or percent decrease, have the students subtract the two values being compared. Then, have them take that answer and divide it by the beginning value to get the ratio of change to starting value. Multiply this value by 100 to get the percent change. For example, the population changed from 1000 to 1150. What was the percent increase? The students would subtract 1150 - 1000 to get 150. They then need to divide 150 by 1000 to get 0.15. Multiply this value by 100 (change a decimal to a percent), and the percent increase is 0.15 × 100 which equals 15%. This  website provides a calculator that will do this.
• The scale on a map suggests that 1 centimeter represents an actual distance of 5 kilometers. The map distance between two towns is 8 centimeters. What is the actual distance? In this situation, a table can help highlight this relationship.

Drawing pictures may help students see that the rate is a scalar concept.

• To help students learn the difference between compound and simple interest, work on simple interest first. In simple interest, the amount of interest earned is proportional to the number of months invested. For example, a deposit of $500 in an account earns 1% simple interest each month. After 1 month, $5 of interest would be earned, because $500×1% = $5. ($500x0.01 = $5).  After two months, there will be $10 of interest earned. Interest is only earned on the original deposit with simple interest. With compound interest, interest is not only calculated on the original deposit, but also the interest that has previously been earned. So, one month, the interest earned would be $500x1% = $5. The balance after that first month is $505. After two months, the interest is earned on the total balance of $505. As shown in the table, the amount added each month is not constant, therefore, compound interest earned is not an example of a proportional relationship, whereas simple interest earned is showing a proportional relationship.

• Proportion problems This website will explain proportions, provide examples, and provides sample problems.
• Ratios and proportions This site includes ratios, comparing ratios, and proportions.
• Ratio and proportion factsheets
• Ratios and proportions in everyday life This site addresses ratios and proportions and how knowledge of these mathematical concepts is used in everyday life. It includes lesson plans, animation, online and printable worksheets, online exercises, games, quizzes, and a link to eThemes Resource on Math: Equivalent Ratios. 
• Real-world proportions This site includes problems with solving proportions (algorithms) and real-world applications.
• Math in daily life: Cooking by numbers This webpage that helps make a connection between proportions and the real-life situation of cooking.
• Burns, M., and Sheffield, S. (2004). Jim and the Beanstalk. In Math and Literature (p. 60) . Sausalito, CA: Math Solutions Publications.
• Burns, M., and Sheffield, S. (2004). How Big is a Foot. In Math and Literature (p. 47). Sausalito, CA: Math Solutions Publications.

• interest: fee paid on loans or earned on invested money, based on the principal amount and the interest rate.
• simple interest: interest paid only on the original principal, not on the interest accrued.
• compound interest: interest computed on accumulated interest as well as on the principal.
• proportion: an equation which states that two ratios are equal; a relationship between two ratios. Example: $\frac{\text{hours spent on homework}}{\text{hours spent in school}}=\frac{2}{7}$

Note that this does not necessarily imply that "hours spent on homework" = 2 or that "hours spent in school" = 7. During a week, 10 hours may have been spent on homework while 35 hours were spent in school. The proportion is still true because $\frac{10}{35}=\frac{2}{7}$.

• proportional reasoning: a mathematical way of thinking in which students recognize proportional versus non-proportional situations and can use multiple approaches, not just the cross-products approach, for solving problems about proportional situations.

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

• What conclusions can be drawn about the student's understanding of applying relationships to solve problems in various contexts?
• Can students scale values up or down and understand why they are doing it?
• Can students tell if a relationship is proportional by looking at the verbal description of it, or do they need to mathematically figure it out?
• Are students able to transfer prior knowledge about equivalent fractions to the concept of proportionality?
• Do students understand why their procedures work?
• What connections have been made as students explored the mathematical characteristics of proportional situations?
• What models would help students in understanding the concepts addressed in the lesson?
• What aspects of proportional relationships are students still struggling with?

Cramer, K. & Post, T. (1993, February). Making connections: A case for proportionality. In Arithmetic Teacher, 60(6), 342-346.

• Massachusetts Comprehensive Assessment System Spring 2010 Test Items http://www.doe.mass.edu/mcas/2010/release/g7math.pdf
• Absolute value http :// www . purplemath . com / modules / absolute . htm
• Adding and subtracting negative numbers http :// www . purplemath . com / modules / negative 2. htm
• Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Philips, E. (2009). Accentuate the Negative, CMP2. Pearson Prentice Hall.
• Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Philips, E. (2009). Comparing and Scaling, CMP2. Pearson Prentice Hall.
• Rational Number Project: Proportional reasoning: the effect of two context variables, rate type, and problem setting http :// www . cehd . umn . edu / rationalnumberproject /89_6. html
• Dacey, L.S., and Gartland, K. (2009). Math for All: Differentiating Instruction. Sausalito, CA: Math Solutions.

Answer: a DOK: Level 3 Source: Minnesota Grade 7 Mathematics MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.4

Answer: c DOK: Level 3 Source: Massachusetts Comprehensive Assessment System Release of Spring 2009 Test Items

Assume you borrow $900 at 7% annual compound interest for four years.

• How much money do you owe at the end of the four years? Show or explain your work.
• What is the total interest you will have to pay? Show or explain your work.

Answers: Part A: $900 + $252 = $1179.72; Part B: $279.72 DOK: Level 2 Source: Test Prep: Modified from MCA III Test Preparation Grade 7, Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 South Park Center Loop, Orlando, FL 32819.

Tim is mixing 1 L of juice concentrate with 5L of water to make juice for his 10 guests. After he pours the mix into 10 different cups, he realizes that the juice is not sweet enough, so he adds 0.1 L of syrup into each of the cups. What is the final amount of juice in each cup?             A. 0.5 L             B. 0.7 L             C. 1.7 L             D. 2.0 L   Answer: b DOK: Level 2 Source: Test Prep: MCA III Test Preparation Grade 7, Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 South Park Center Loop, Orlando, FL 32819.

Answer: b DOK: Level 2 Source: Minnesota Grade 7 Mathematics MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.5

Answer: b DOK: Level 2 Source: Minnesota Grade 7 Mathematics Modified MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.5

Answer: d DOK: Level 2 Source: Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

Answer: b DOK: Level 2 Source: Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

Differentiation

• Provide students with multiplication tables.
• Students may need to be given a place value chart to help them in rounding to the correct place (hundreds vs. hundredths).
• Use pictures and or tables to help the students see the pattern; always label the values they are trying to scale, such as 10 in = 3 ft, 3 in = ________ft. By keeping the labels with the numbers, fewer errors in relationships will be made.
• Students may see the word "ratio" as "radios." Assist them by clarifying the meaning and pronunciation of both words.
• Use a graphic organizer displaying operations with integer rules .
• Introduce √ of a number to plot on a number line.
• Do multi-step conversion-type problems to show proportional relationship.
• Explain the concept of compound interest using exponential growth and exponential equations.
• Create a gameboard activity. Have students work in groups of four to create gameboards marked with mathematical questions they must answer to be able to move ahead on the boards. They should use at least three addition, three subtraction, three multiplication and three division operations. They should also use positive numbers, negative numbers, decimals, and fractions. Students will fill in operation symbols and numbers on the boards. When they are done, the class can play the different games.

Parents/Admin

Administrative/peer classroom observation.

Parent Resources

• Math games, problems and puzzles

Related Frameworks

7.1.2a applying & making sense of rational numbers.

• 7.1.2.1 Arithmetic Procedures
• 7.1.2.2 Explain Arithmetic Procedures
• 7.1.2.3 Calculators & Rational Numbers
• 7.1.2.6 Absolute Value
• 7.1.2.4 Solve Problems with Rational Numbers Including Positive Integer Exponents
• 7.1.2.5 Proportional Reasoning