dividing rational numbers practice and problem solving d

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Integrated math 3

Course: integrated math 3   >   unit 13.

• Multiplying & dividing rational expressions: monomials
• Multiplying rational expressions

Dividing rational expressions

• Multiply & divide rational expressions: Error analysis
• Multiply & divide rational expressions
• Multiplying rational expressions: multiple variables
• Dividing rational expressions: unknown expression
• Multiply & divide rational expressions (advanced)

What you should be familiar with before taking this lesson

• Intro to rational expressions
• Intro to simplifying rational expressions

What you will learn in this lesson

Dividing fractions, example 1: 3 x 4 4 ÷ 9 x 10 ‍  .

• for any value that makes either of the original rational expressions undefined, Why? It's important to remember that the resulting expression did not show up out of nowhere. Instead, it is a result that we created by dividing two rational expressions! If one of these two original rational expressions had been undefined, we would not have been able to find the quotient in the first place.
• and for any value that makes the divisor equal to zero. Why? For all rational expressions, the denominator cannot be equal to zero because division by zero is undefined. For the same reason, in the case of a division by a rational expression , that expression cannot be equal to zero as well! A rational expression is equal to zero when its numerator is equal to zero.
• The dividend 3 x 4 4 ‍   is defined for all x ‍   -values.
• The divisor 9 x 10 ‍   is defined for all x ‍   -values, and is equal to zero for x = 0 ‍   .

5 x 3 6 ‍   for x ≠ 0 ‍  

Check your understanding

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  
• The dividend 3 10 x 2 ‍   is undefined for x = 0 ‍   .
• The divisor 6 15 x 5 ‍   is undefined for x = 0 ‍   .

3 x 3 4 ‍   for x ≠ 0 ‍  

Example 2: x 2 + x − 6 x 2 + 3 x − 10 ÷ x + 3 x − 5 ‍  

• The dividend ( x + 3 ) ( x − 2 ) ( x + 5 ) ( x − 2 ) ‍   is defined for x ≠ − 5 , 2 ‍   .
• The divisor x + 3 x − 5 ‍   is defined for x ≠ 5 ‍   , and is equal to zero for x = − 3 ‍   .

x − 5 x + 5 ‍   for x ≠ 5 , 2 , − 3 ‍  

• (Choice A)   x ≠ 7 ‍   A x ≠ 7 ‍  
• (Choice B)   x ≠ 0 ‍   B x ≠ 0 ‍  
• (Choice C)   x ≠ − 2 ‍   C x ≠ − 2 ‍  
• (Choice D)   x ≠ − 1 ‍   D x ≠ − 1 ‍  
• (Choice E)   x ≠ 2 ‍   E x ≠ 2 ‍  
• The dividend x − 7 ( x − 2 ) ( x + 2 ) ‍   is defined for x ≠ 2 , − 2 ‍   .
• The divisor ( x − 7 ) ( x + 1 ) 2 ( x + 2 ) ‍   is defined for x ≠ − 2 ‍   , and is equal to zero for x = 7 , − 1 ‍   .
• (Choice A)   x ≠ − 3 ‍   A x ≠ − 3 ‍  
• (Choice B)   x ≠ − 4 ‍   B x ≠ − 4 ‍  
• (Choice C)   x ≠ − 1 ‍   C x ≠ − 1 ‍  
• (Choice D)   x ≠ 1 ‍   D x ≠ 1 ‍  
• (Choice E)   x ≠ 3 ‍   E x ≠ 3 ‍  
• The dividend x + 4 ( x + 3 ) ( x − 3 ) ‍   is defined for x ≠ − 3 , 3 ‍   .
• The divisor x − 1 ( x − 3 ) ( x − 1 ) ‍   is defined for x ≠ 3 , 1 ‍   .

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7.1 Multiply and Divide Rational Expressions

Learning objectives.

By the end of this section, you will be able to:

• Determine the values for which a rational expression is undefined
• Simplify rational expressions
• Multiply rational expressions
• Divide rational expressions
• Multiply and divide rational functions

Be Prepared 7.1

Before you get started, take this readiness quiz.

Simplify: 90 y 15 y 2 . 90 y 15 y 2 . If you missed this problem, review Example 5.13 .

Be Prepared 7.2

Multiply: 14 15 · 6 35 . 14 15 · 6 35 . If you missed this problem, review Example 1.25 .

Be Prepared 7.3

Divide: 12 10 ÷ 8 25 . 12 10 ÷ 8 25 . If you missed this problem, review Example 1.26 .

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression .

Rational Expression

A rational expression is an expression of the form p q , p q , where p and q are polynomials and q ≠ 0 . q ≠ 0 .

Here are some examples of rational expressions:

Notice that the first rational expression listed above, − 24 56 − 24 56 , is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

Determine the Values for Which a Rational Expression is Undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

Determine the values for which a rational expression is undefined.

• Step 1. Set the denominator equal to zero.
• Step 2. Solve the equation.

Example 7.1

Determine the value for which each rational expression is undefined:

ⓐ 8 a 2 b 3 c 8 a 2 b 3 c ⓑ 4 b − 3 2 b + 5 4 b − 3 2 b + 5 ⓒ x + 4 x 2 + 5 x + 6 . x + 4 x 2 + 5 x + 6 .

The expression will be undefined when the denominator is zero.

Determine the value for which each rational expression is undefined.

ⓐ 3 y 2 8 x 3 y 2 8 x ⓑ 8 n − 5 3 n + 1 8 n − 5 3 n + 1 ⓒ a + 10 a 2 + 4 a + 3 a + 10 a 2 + 4 a + 3

ⓐ 4 p 5 q 4 p 5 q ⓑ y − 1 3 y + 2 y − 1 3 y + 2 ⓒ m − 5 m 2 + m − 6 m − 5 m 2 + m − 6

Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.

Simplified Rational Expression

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example,

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

Equivalent Fractions Property

If a , b , and c are numbers where b ≠ 0 , c ≠ 0 , b ≠ 0 , c ≠ 0 ,

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see b ≠ 0 , c ≠ 0 b ≠ 0 , c ≠ 0 clearly stated.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

Removing the x ’s from x + 5 x x + 5 x would be like cancelling the 2’s in the fraction 2 + 5 2 ! 2 + 5 2 !

Example 7.2

How to simplify a rational expression.

Simplify: x 2 + 5 x + 6 x 2 + 8 x + 12 x 2 + 5 x + 6 x 2 + 8 x + 12 .

Simplify: x 2 − x − 2 x 2 − 3 x + 2 . x 2 − x − 2 x 2 − 3 x + 2 .

Simplify: x 2 − 3 x − 10 x 2 + x − 2 . x 2 − 3 x − 10 x 2 + x − 2 .

We now summarize the steps you should follow to simplify rational expressions.

Simplify a rational expression.

• Step 1. Factor the numerator and denominator completely.
• Step 2. Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.

We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.

Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Example 7.3

Simplify: 3 a 2 − 12 a b + 12 b 2 6 a 2 − 24 b 2 3 a 2 − 12 a b + 12 b 2 6 a 2 − 24 b 2 .

Simplify: 2 x 2 − 12 x y + 18 y 2 3 x 2 − 27 y 2 2 x 2 − 12 x y + 18 y 2 3 x 2 − 27 y 2 .

Simplify: 5 x 2 − 30 x y + 25 y 2 2 x 2 − 50 y 2 5 x 2 − 30 x y + 25 y 2 2 x 2 − 50 y 2 .

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is − a − a and − a = −1 · a . − a = −1 · a .

The numerical fraction, say 7 −7 7 −7 simplifies to −1 −1 . We also recognize that the numerator and denominator are opposites.

The fraction a − a a − a , whose numerator and denominator are opposites also simplifies to −1 −1 .

This tells us that b − a b − a is the opposite of a − b . a − b .

In general, we could write the opposite of a − b a − b as b − a . b − a . So the rational expression a − b b − a a − b b − a simplifies to −1 . −1 .

Opposites in a Rational Expression

The opposite of a − b a − b is b − a . b − a .

An expression and its opposite divide to −1 . −1 .

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat a + b a + b and b + a b + a as opposites. Recall that in addition, order doesn’t matter so a + b = b + a a + b = b + a . So if a ≠ − b a ≠ − b , then a + b b + a = 1 . a + b b + a = 1 .

Example 7.4

Simplify: x 2 − 4 x − 32 64 − x 2 . x 2 − 4 x − 32 64 − x 2 .

Simplify: x 2 − 4 x − 5 25 − x 2 . x 2 − 4 x − 5 25 − x 2 .

Simplify: x 2 + x − 2 1 − x 2 . x 2 + x − 2 1 − x 2 .

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Multiplication of Rational Expressions

If p , q , r , and s are polynomials where q ≠ 0 , s ≠ 0 , q ≠ 0 , s ≠ 0 , then

To multiply rational expressions, multiply the numerators and multiply the denominators.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x ≠ 0 , x ≠ 0 , x ≠ 3 , x ≠ 3 , and x ≠ 4 . x ≠ 4 .

Example 7.5

How to multiply rational expressions.

Simplify: 2 x x 2 − 7 x + 12 · x 2 − 9 6 x 2 . 2 x x 2 − 7 x + 12 · x 2 − 9 6 x 2 .

Simplify: 5 x x 2 + 5 x + 6 · x 2 − 4 10 x . 5 x x 2 + 5 x + 6 · x 2 − 4 10 x .

Try It 7.10

Simplify: 9 x 2 x 2 + 11 x + 30 · x 2 − 36 3 x 2 . 9 x 2 x 2 + 11 x + 30 · x 2 − 36 3 x 2 .

Multiply rational expressions.

• Step 1. Factor each numerator and denominator completely.
• Step 2. Multiply the numerators and denominators.
• Step 3. Simplify by dividing out common factors.

Example 7.6

Multiply: 3 a 2 − 8 a − 3 a 2 − 25 · a 2 + 10 a + 25 3 a 2 − 14 a − 5 . 3 a 2 − 8 a − 3 a 2 − 25 · a 2 + 10 a + 25 3 a 2 − 14 a − 5 .

Try It 7.11

Simplify: 2 x 2 + 5 x − 12 x 2 − 16 · x 2 − 8 x + 16 2 x 2 − 13 x + 15 . 2 x 2 + 5 x − 12 x 2 − 16 · x 2 − 8 x + 16 2 x 2 − 13 x + 15 .

Try It 7.12

Simplify: 4 b 2 + 7 b − 2 1 − b 2 · b 2 − 2 b + 1 4 b 2 + 15 b − 4 . 4 b 2 + 7 b − 2 1 − b 2 · b 2 − 2 b + 1 4 b 2 + 15 b − 4 .

Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.

Division of Rational Expressions

If p , q , r, and s are polynomials where q ≠ 0 , r ≠ 0 , s ≠ 0 , q ≠ 0 , r ≠ 0 , s ≠ 0 , then

To divide rational expressions, multiply the first fraction by the reciprocal of the second.

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.

Example 7.7

How to divide rational expressions.

Divide: p 3 + q 3 2 p 2 + 2 p q + 2 q 2 ÷ p 2 − q 2 6 . p 3 + q 3 2 p 2 + 2 p q + 2 q 2 ÷ p 2 − q 2 6 .

Try It 7.13

Simplify: x 3 + 8 3 x 2 − 6 x + 12 ÷ x 2 − 4 6 . x 3 + 8 3 x 2 − 6 x + 12 ÷ x 2 − 4 6 .

Try It 7.14

Simplify: 2 z 2 z 2 − 1 ÷ z 3 − z 2 + z z 3 + 1 . 2 z 2 z 2 − 1 ÷ z 3 − z 2 + z z 3 + 1 .

Divide rational expressions.

• Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
• Step 2. Factor the numerators and denominators completely.
• Step 3. Multiply the numerators and denominators together.
• Step 4. Simplify by dividing out common factors.

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

Example 7.8

Divide: 6 x 2 − 7 x + 2 4 x − 8 2 x 2 − 7 x + 3 x 2 − 5 x + 6 . 6 x 2 − 7 x + 2 4 x − 8 2 x 2 − 7 x + 3 x 2 − 5 x + 6 .

Try It 7.15

Simplify: 3 x 2 + 7 x + 2 4 x + 24 3 x 2 − 14 x − 5 x 2 + x − 30 . 3 x 2 + 7 x + 2 4 x + 24 3 x 2 − 14 x − 5 x 2 + x − 30 .

Try It 7.16

Simplify: y 2 − 36 2 y 2 + 11 y − 6 2 y 2 − 2 y − 60 8 y − 4 . y 2 − 36 2 y 2 + 11 y − 6 2 y 2 − 2 y − 60 8 y − 4 .

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

Example 7.9

Perform the indicated operations: 3 x − 6 4 x − 4 · x 2 + 2 x − 3 x 2 − 3 x − 10 ÷ 2 x + 12 8 x + 16 . 3 x − 6 4 x − 4 · x 2 + 2 x − 3 x 2 − 3 x − 10 ÷ 2 x + 12 8 x + 16 .

Try It 7.17

Perform the indicated operations: 4 m + 4 3 m − 15 · m 2 − 3 m − 10 m 2 − 4 m − 32 ÷ 12 m − 36 6 m − 48 . 4 m + 4 3 m − 15 · m 2 − 3 m − 10 m 2 − 4 m − 32 ÷ 12 m − 36 6 m − 48 .

Try It 7.18

Perform the indicated operations: 2 n 2 + 10 n n − 1 ÷ n 2 + 10 n + 24 n 2 + 8 n − 9 · n + 4 8 n 2 + 12 n . 2 n 2 + 10 n n − 1 ÷ n 2 + 10 n + 24 n 2 + 8 n − 9 · n + 4 8 n 2 + 12 n .

Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form p q , p q , where p and q are polynomials and q ≠ 0 . q ≠ 0 . Similarly, we define a rational function as a function of the form R ( x ) = p ( x ) q ( x ) R ( x ) = p ( x ) q ( x ) where p ( x ) p ( x ) and q ( x ) q ( x ) are polynomial functions and q ( x ) q ( x ) is not zero.

Rational Function

A rational function is a function of the form

where p ( x ) p ( x ) and q ( x ) q ( x ) are polynomial functions and q ( x ) q ( x ) is not zero.

The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make q ( x ) = 0 . q ( x ) = 0 .

Determine the domain of a rational function.

• Step 3. The domain is all real numbers excluding the values found in Step 2.

Example 7.10

Find the domain of R ( x ) = 2 x 2 − 14 x 4 x 2 − 16 x − 48 . R ( x ) = 2 x 2 − 14 x 4 x 2 − 16 x − 48 .

The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.

Try It 7.19

Find the domain of R ( x ) = 2 x 2 − 10 x 4 x 2 − 16 x − 20 . R ( x ) = 2 x 2 − 10 x 4 x 2 − 16 x − 20 .

Try It 7.20

Find the domain of R ( x ) = 4 x 2 − 16 x 8 x 2 − 16 x − 64 . R ( x ) = 4 x 2 − 16 x 8 x 2 − 16 x − 64 .

To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.

Example 7.11

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = 2 x − 6 x 2 − 8 x + 15 f ( x ) = 2 x − 6 x 2 − 8 x + 15 and g ( x ) = x 2 − 25 2 x + 10 . g ( x ) = x 2 − 25 2 x + 10 .

Try It 7.21

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = 3 x − 21 x 2 − 9 x + 14 f ( x ) = 3 x − 21 x 2 − 9 x + 14 and g ( x ) = 2 x 2 − 8 3 x + 6 . g ( x ) = 2 x 2 − 8 3 x + 6 .

Try It 7.22

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = x 2 − x 3 x 2 + 27 x − 30 f ( x ) = x 2 − x 3 x 2 + 27 x − 30 and g ( x ) = x 2 − 100 x 2 − 10 x . g ( x ) = x 2 − 100 x 2 − 10 x .

To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.

Example 7.12

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 3 x 2 x 2 − 4 x f ( x ) = 3 x 2 x 2 − 4 x and g ( x ) = 9 x 2 − 45 x x 2 − 7 x + 10 . g ( x ) = 9 x 2 − 45 x x 2 − 7 x + 10 .

Try It 7.23

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 2 x 2 x 2 − 8 x f ( x ) = 2 x 2 x 2 − 8 x and g ( x ) = 8 x 2 + 24 x x 2 + x − 6 . g ( x ) = 8 x 2 + 24 x x 2 + x − 6 .

Try It 7.24

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 15 x 2 3 x 2 + 33 x f ( x ) = 15 x 2 3 x 2 + 33 x and g ( x ) = 5 x − 5 x 2 + 9 x − 22 . g ( x ) = 5 x − 5 x 2 + 9 x − 22 .

Section 7.1 Exercises

Practice makes perfect.

In the following exercises, determine the values for which the rational expression is undefined.

ⓐ 2 x 2 z 2 x 2 z , ⓑ 4 p − 1 6 p − 5 4 p − 1 6 p − 5 , ⓒ n − 3 n 2 + 2 n − 8 n − 3 n 2 + 2 n − 8

ⓐ 10 m 11 n 10 m 11 n , ⓑ 6 y + 13 4 y − 9 6 y + 13 4 y − 9 , ⓒ b − 8 b 2 − 36 b − 8 b 2 − 36

ⓐ 4 x 2 y 3 y 4 x 2 y 3 y , ⓑ 3 x − 2 2 x + 1 3 x − 2 2 x + 1 , ⓒ u − 1 u 2 − 3 u − 28 u − 1 u 2 − 3 u − 28

ⓐ 5 p q 2 9 q 5 p q 2 9 q , ⓑ 7 a − 4 3 a + 5 7 a − 4 3 a + 5 , ⓒ 1 x 2 − 4 1 x 2 − 4

In the following exercises, simplify each rational expression.

− 44 55 − 44 55

56 63 56 63

8 m 3 n 12 m n 2 8 m 3 n 12 m n 2

36 v 3 w 2 27 v w 3 36 v 3 w 2 27 v w 3

8 n − 96 3 n − 36 8 n − 96 3 n − 36

12 p − 240 5 p − 100 12 p − 240 5 p − 100

x 2 + 4 x − 5 x 2 − 2 x + 1 x 2 + 4 x − 5 x 2 − 2 x + 1

y 2 + 3 y − 4 y 2 − 6 y + 5 y 2 + 3 y − 4 y 2 − 6 y + 5

a 2 − 4 a 2 + 6 a − 16 a 2 − 4 a 2 + 6 a − 16

y 2 − 2 y − 3 y 2 − 9 y 2 − 2 y − 3 y 2 − 9

p 3 + 3 p 2 + 4 p + 12 p 2 + p − 6 p 3 + 3 p 2 + 4 p + 12 p 2 + p − 6

x 3 − 2 x 2 − 25 x + 50 x 2 − 25 x 3 − 2 x 2 − 25 x + 50 x 2 − 25

8 b 2 − 32 b 2 b 2 − 6 b − 80 8 b 2 − 32 b 2 b 2 − 6 b − 80

−5 c 2 − 10 c −10 c 2 + 30 c + 100 −5 c 2 − 10 c −10 c 2 + 30 c + 100

3 m 2 + 30 m n + 75 n 2 4 m 2 − 100 n 2 3 m 2 + 30 m n + 75 n 2 4 m 2 − 100 n 2

5 r 2 + 30 r s − 35 s 2 r 2 − 49 s 2 5 r 2 + 30 r s − 35 s 2 r 2 − 49 s 2

a − 5 5 − a a − 5 5 − a

5 − d d − 5 5 − d d − 5

20 − 5 y y 2 − 16 20 − 5 y y 2 − 16

4 v − 32 64 − v 2 4 v − 32 64 − v 2

w 3 + 216 w 2 − 36 w 3 + 216 w 2 − 36

v 3 + 125 v 2 − 25 v 3 + 125 v 2 − 25

z 2 − 9 z + 20 16 − z 2 z 2 − 9 z + 20 16 − z 2

a 2 − 5 a − 36 81 − a 2 a 2 − 5 a − 36 81 − a 2

In the following exercises, multiply the rational expressions.

12 16 · 4 10 12 16 · 4 10

32 5 · 16 24 32 5 · 16 24

5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2 5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2

12 a 3 b b 2 · 2 a b 2 9 b 3 12 a 3 b b 2 · 2 a b 2 9 b 3

5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p 5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p

3 q 2 q 2 + q − 6 · q 2 − 9 9 q 3 q 2 q 2 + q − 6 · q 2 − 9 9 q

2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y 2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y

z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2 z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2

28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49 28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49

72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36 72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36

3 c 2 − 16 c + 5 c 2 − 25 · c 2 + 10 c + 25 3 c 2 − 14 c − 5 3 c 2 − 16 c + 5 c 2 − 25 · c 2 + 10 c + 25 3 c 2 − 14 c − 5

2 d 2 + d − 3 d 2 − 16 · d 2 − 8 d + 16 2 d 2 − 9 d − 18 2 d 2 + d − 3 d 2 − 16 · d 2 − 8 d + 16 2 d 2 − 9 d − 18

6 m 2 − 13 m + 2 9 − m 2 · m 2 − 6 m + 9 6 m 2 + 23 m − 4 6 m 2 − 13 m + 2 9 − m 2 · m 2 − 6 m + 9 6 m 2 + 23 m − 4

2 n 2 − 3 n − 14 25 − n 2 · n 2 − 10 n + 25 2 n 2 − 13 n + 21 2 n 2 − 3 n − 14 25 − n 2 · n 2 − 10 n + 25 2 n 2 − 13 n + 21

In the following exercises, divide the rational expressions.

v − 5 11 − v ÷ v 2 − 25 v − 11 v − 5 11 − v ÷ v 2 − 25 v − 11

10 + w w − 8 ÷ 100 − w 2 8 − w 10 + w w − 8 ÷ 100 − w 2 8 − w

3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64 3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64

r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45 r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45

p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12 p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12

v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4 v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4

x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 ) x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 )

2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z ) 2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z )

2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16 2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16

3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15 3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15

12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5 12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5

4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4 4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4

For the following exercises, perform the indicated operations.

10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 ÷ 5 m 2 + 10 m 2 m − 10 10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 ÷ 5 m 2 + 10 m 2 m − 10

4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 ÷ 108 n 2 − 24 n n + 6 4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 ÷ 108 n 2 − 24 n n + 6

12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 · p − 7 9 p 3 − 9 p 2 12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 · p − 7 9 p 3 − 9 p 2

6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 · 4 q 2 + 12 q 12 q + 6 6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 · 4 q 2 + 12 q 12 q + 6

In the following exercises, find the domain of each function.

R ( x ) = x 3 − 2 x 2 − 25 x + 50 x 2 − 25 R ( x ) = x 3 − 2 x 2 − 25 x + 50 x 2 − 25

R ( x ) = x 3 + 3 x 2 − 4 x − 12 x 2 − 4 R ( x ) = x 3 + 3 x 2 − 4 x − 12 x 2 − 4

R ( x ) = 3 x 2 + 15 x 6 x 2 + 6 x − 36 R ( x ) = 3 x 2 + 15 x 6 x 2 + 6 x − 36

R ( x ) = 8 x 2 − 32 x 2 x 2 − 6 x − 80 R ( x ) = 8 x 2 − 32 x 2 x 2 − 6 x − 80

For the following exercises, find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) f ( x ) and g ( x ) g ( x ) are given.

f ( x ) = 6 x 2 − 12 x x 2 + 7 x − 18 f ( x ) = 6 x 2 − 12 x x 2 + 7 x − 18 g ( x ) = x 2 − 81 3 x 2 − 27 x g ( x ) = x 2 − 81 3 x 2 − 27 x

f ( x ) = x 2 − 2 x x 2 + 6 x − 16 f ( x ) = x 2 − 2 x x 2 + 6 x − 16 g ( x ) = x 2 − 64 x 2 − 8 x g ( x ) = x 2 − 64 x 2 − 8 x

f ( x ) = 4 x x 2 − 3 x − 10 f ( x ) = 4 x x 2 − 3 x − 10 g ( x ) = x 2 − 25 8 x 2 g ( x ) = x 2 − 25 8 x 2

f ( x ) = 2 x 2 + 8 x x 2 − 9 x + 20 f ( x ) = 2 x 2 + 8 x x 2 − 9 x + 20 g ( x ) = x − 5 x 2 g ( x ) = x − 5 x 2

For the following exercises, find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) f ( x ) and g ( x ) g ( x ) are given.

f ( x ) = 27 x 2 3 x − 21 f ( x ) = 27 x 2 3 x − 21 g ( x ) = 3 x 2 + 18 x x 2 + 13 x + 42 g ( x ) = 3 x 2 + 18 x x 2 + 13 x + 42

f ( x ) = 24 x 2 2 x − 8 f ( x ) = 24 x 2 2 x − 8 g ( x ) = 4 x 3 + 28 x 2 x 2 + 11 x + 28 g ( x ) = 4 x 3 + 28 x 2 x 2 + 11 x + 28

f ( x ) = 16 x 2 4 x + 36 f ( x ) = 16 x 2 4 x + 36 g ( x ) = 4 x 2 − 24 x x 2 + 4 x − 45 g ( x ) = 4 x 2 − 24 x x 2 + 4 x − 45

f ( x ) = 24 x 2 2 x − 4 f ( x ) = 24 x 2 2 x − 4 g ( x ) = 12 x 2 + 36 x x 2 − 11 x + 18 g ( x ) = 12 x 2 + 36 x x 2 − 11 x + 18

Writing Exercises

Explain how you find the values of x for which the rational expression x 2 − x − 20 x 2 − 4 x 2 − x − 20 x 2 − 4 is undefined.

Explain all the steps you take to simplify the rational expression p 2 + 4 p − 21 9 − p 2 . p 2 + 4 p − 21 9 − p 2 .

ⓐ Multiply 7 4 · 9 10 7 4 · 9 10 and explain all your steps. ⓑ Multiply n n − 3 · 9 n + 3 n n − 3 · 9 n + 3 and explain all your steps. ⓒ Evaluate your answer to part ⓑ when n = 7 n = 7 . Did you get the same answer you got in part ⓐ ? Why or why not?

ⓐ Divide 24 5 ÷ 6 24 5 ÷ 6 and explain all your steps. ⓑ Divide x 2 − 1 x ÷ ( x + 1 ) x 2 − 1 x ÷ ( x + 1 ) and explain all your steps. ⓒ Evaluate your answer to part ⓑ when x = 5 . x = 5 . Did you get the same answer you got in part ⓐ ? Why or why not?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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Dividing Rational Numbers Lesson Plan

Get the lesson materials.

Dividing Rational Numbers Fractions Decimals Guided Notes Sketch & Doodles

Ever wondered how to teach dividing rational numbers, including fractions, integers, and decimals, in an engaging way to your middle school students?

In this lesson plan, students will learn about dividing rational numbers and their real-life applications. Through artistic and interactive guided notes, check for understanding questions, a color by code activity, and a maze worksheet, students will gain a comprehensive understanding of dividing rational numbers.

The lesson culminates with a real-life example that explores how dividing rational numbers can be applied to splitting a bill at a restaurant.

• Standards : CCSS 7.NS.A.2 , CCSS 7.NS.A.2.a , CCSS 7.NS.A.2.c
• Topics : Integers & Rational Numbers , Fractions , Decimals
• Grade : 7th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Divide rational numbers, including fractions, integers, and decimals

Solve division problems involving positive and negative rational numbers

Apply division of rational numbers to real-life situations

Prerequisites

Before this lesson, students should be familiar with:

Basic operations with rational numbers (adding, subtracting, and multiplying)

Basic understanding of fractions and decimals

Knowledge of how to determine the greatest common factor (GCF) and least common multiple (LCM) of numbers

Colored pencils or markers

Dividing Rational Numbers Fractions Decimals Guided Notes

Key Vocabulary

Rational numbers

Introduction

As a hook, ask students why dividing rational numbers, including fractions, integers, and decimals, is important in real life. Refer to the real-life math application on the last page of the guided notes as well as the FAQs below for ideas.

Use the guided notes to introduce the concept of dividing rational numbers. Walk through the key points of the topic, including the steps and techniques involved in dividing rational numbers. Refer to the FAQ below for a walk through on this, as well as ideas on how to respond to common student questions.

Check for Understanding : Have students walk through the "You Try!" section of the guided notes. Call on students to talk through their answers, potentially on the whiteboard or projector. Based on student responses, reteach concepts that students need extra help with.

Based on student responses, reteach concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice dividing rational numbers including fractions, integers, and decimals using the color by code activity included in the resource. Walk around the classroom to answer any student questions and provide assistance as needed.

Fast finishers can work on the maze activity for extra practice. You can assign these activities as homework for the remainder of the class.

Real-Life Application

Bring the class back together, and introduce the concept of rational number division applied to splitting a bill with friends. Refer to the FAQ for more real life applications that you can use for the discussion!

Additional Self-Checking Digital Practice

If you’re looking for digital practice for dividing rational numbers, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here's an activity to try:

Multiplying & Dividing Rational Numbers Digital Pixel Art

Additional Print Practice

A fun, no-prep way to practice dividing rational numbers is Doodle Math — they’re a fresh take on color by number or color by code. It includes multiple levels of practice, perfect for a review day or sub plan.

Multiplying & Dividing Rational Numbers | Doodle Math: Twist on Color by Number

What is dividing rational numbers? Open

Dividing rational numbers involves dividing numbers that can be expressed as fractions or decimals. It is the process of finding how many times one number can be evenly divided by another number.

How do you divide fractions? Open

To divide fractions, you multiply the first fraction by the reciprocal (flipped) form of the second fraction. This can be done by multiplying the numerators together and the denominators together. Simplify the resulting fraction if possible.

How do you divide decimals? Open

Dividing decimals is similar to dividing whole numbers. Use long division to divide the decimal dividend by the decimal divisor. Place the decimal point in the quotient directly above the decimal point in the dividend.

Can you divide positive and negative rational numbers? Open

Yes, you can divide positive and negative rational numbers. The rules for dividing positive and negative numbers are the same as for multiplying them. The result of the division will have a positive quotient if both numbers have the same sign, and a negative quotient if the numbers have different signs.

What is the difference between dividing fractions and dividing decimals? Open

The main difference is in the representation of the numbers. Dividing fractions involves dividing numbers expressed as fractions, while dividing decimals involves dividing numbers expressed as decimal numbers. The processes and calculations are similar, but the final answers may be in different forms.

How can dividing rational numbers be applied in real life? Open

Dividing rational numbers is commonly used in real-life situations such as dividing the bill for a pizza among friends, calculating the cost per unit of a product, or determining the average speed of a moving object. It helps in solving problems that involve sharing, distributing, or comparing quantities.

Are there any tips or tricks for dividing rational numbers? Open

One tip for dividing rational numbers is to always simplify the fraction before dividing. This makes the calculation easier and reduces the chances of errors. Additionally, keeping track of the signs (+/-) and placing the decimal point correctly when dividing decimals will help in obtaining accurate results.

What are some common mistakes to avoid when dividing rational numbers? Open

Common mistakes to avoid when dividing rational numbers include forgetting to simplify the fraction, reversing the order of the fractions when finding the reciprocal, misplacing the decimal point when dividing decimals, and forgetting to consider the signs of the numbers being divided.

Are there any resources available to practice dividing rational numbers? Open

Yes, there are various resources available for practicing dividing rational numbers. This lesson plan includes guided notes, practice worksheets, color by code activities, and a real-life math application.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

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Common Core: 7th Grade Math : Divide Rational Numbers

Study concepts, example questions & explanations for common core: 7th grade math, all common core: 7th grade math resources, example questions, example question #1 : divide rational numbers.

We know the following information:

• A negative number divided by a positive number will always equal a negative number, and a positive number divided by a negative number will always equal a negative number.
• A negative number divided by a negative number will always equal a positive number 

Example Question #2 : Divide Rational Numbers

Example Question #3 : Divide Rational Numbers

In this particular case, do the negative numbers change our answer? . There are a couple of rules that we need to remember when multiplying with negative numbers:

Example Question #4 : Divide Rational Numbers

Example Question #6 : Divide Rational Numbers

Solve the following: 

Report an issue with this question

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5.3: Divide Rational Expressions

• Last updated
• Save as PDF
• Page ID 83138

• Jennifer Freidenreich
• Diablo Valley College

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Dividing rational expressions is very much the same as fraction division in arithmetic. The first step is to change the division to multiplication and take the reciprocal of the second fraction. Two examples are shown below. Compare the similarities!

\(\begin{array} &&\text{Divide rational numbers} && \text{Divide rational expressions} \\ &\dfrac{3}{4} ÷ \dfrac{9}{20} && \dfrac{x - 6}{x^2 - 16} ÷ \dfrac{4x-24}{x^2 + x - 12} \\ &\dfrac{3}{4} \cdot \dfrac{20}{9} &\textcolor{green}{\text{Multiply by the reciprocal}} & \dfrac{x - 6}{x^2 - 16} \cdot \dfrac{x^2 + x - 12}{4x-24} \\ &=\dfrac{3}{4} \cdot \dfrac{4 \cdot 5}{3 \cdot 3} &\textcolor{green}{\text{Factor and cancel}} & \dfrac{x - 6}{(x-4)(x+4)} \cdot \dfrac{(x+4)(x-3)}{4(x-6)} \\ &\dfrac{5}{3} &\textcolor{green}{\text{What's left?}} & \dfrac{x-3}{4(x-4)} \\ &\text{This answer is in simplest form.} &\textcolor{green}{\text{Simplify answer}}& \dfrac{x-3}{4x-16} \end{array}\)

Complex Fractions

There are two dueling notations for dividing rational expressions. The above notation uses the symbol \(÷\) for division. The competing notation is called a complex fraction, which maintains fraction notation. A horizontal fraction bar indicates division.

A complex fraction is a fraction in which either the numerator is a fraction, or the denominator is a fraction, or both. To simplify complex fractions, translate the main fraction bar to division.

\(\begin{array} &\dfrac{\frac{a}{b}}{\frac{c}{d}} &= \dfrac{a}{b} ÷ \dfrac{c}{d} &\text{Step \(1\): Translate to \(÷\) notation.} \\ &= \dfrac{a}{b} \cdot \dfrac{d}{c} &\text{Step \(2\): Multiply by the reciprocal.} \\ &= \dfrac{ad}{bc} &\text{Step \(3\): Multiply and simplify.} \end{array}\)

Example 5.3.1

Simplify \(\dfrac{\frac{8u^3}{35w^4}}{\frac{24u}{5w^3}}\)

Do you see the main fraction bar? \(\dfrac{\frac{8u^3}{35w^4}}{\frac{24u}{5w^3}}\) \(\textcolor{blue}{\longleftarrow \text{Change fraction bar to } ÷} \)

\( \dfrac{8u^3}{35w^4} ÷\dfrac{24u}{5w^3} = \dfrac{\cancel{8u^3}^{\textcolor{red}{u^2}}}{\cancel{35w^4}^{\textcolor{red}{7w}}} \cdot \dfrac{\cancel{5w^3}^{\textcolor{red}{1}}}{\cancel{24u}^{\textcolor{red}{3}}} = \dfrac{u^2}{21w} \)

Example 5.3.2

Simplify \(\dfrac{\frac{2}{h}}{10h}\)

Let’s convert \(10h\) to fraction form: \(\dfrac{\frac{2}{h}}{\frac{10h}{1}}\) \(\textcolor{blue}{\longleftarrow \text{Change fraction bar to } ÷} \)

\( \dfrac{2}{h} ÷\dfrac{10h}{1} = \dfrac{\cancel{2}^{\textcolor{red}{1}}}{h} \cdot \dfrac{1}{\cancel{10h}^{\textcolor{red}{5h}}} = \dfrac{1}{5h^2} \)

Example 5.3.3

Simplify \(\dfrac{\frac{t^2-1}{t+2}}{\frac{5-5t}{t^2+3t+2}}\)

The given expression is a complex fraction.

\(\begin{array}&\dfrac{\frac{t^2-1}{t+2}}{\frac{5-5t}{t^2+3t+2}} &= \dfrac{t^2-1}{t+2} ÷ \dfrac{5-5t}{t^2+3t+2} &\text{Change the main fraction bar to \(÷\)} \\ &= \dfrac{t^2-1}{t+2} \cdot \dfrac{t^2+3t+2}{5-5t} &\text{Multiply by the reciprocal.} \\ &= \dfrac{(t+1)(t-1)}{(t+2)} \cdot \dfrac{(t+2)(t+1)}{5(1-t)} = \dfrac{(t+1) (\cancel{t-1})}{(\cancel{t+2})} \cdot \dfrac{(\cancel{t+2})(t+1)}{5(\cancel{1-t})^{\textcolor{red}{-1}}} &\text{Factor and cancel. Opposites, too!} \\ &= −\dfrac{(t+1)^2}{5} & \end{array}\)

In our final example (below), we divide three rational expressions. As a student, allow yourself to apply your knowledge of fractions. The process is the same for rational expressions. Before doing example 4, you might try to simplify the following: \(\dfrac{1}{2} ÷ \dfrac{3}{4} ÷ \dfrac{3}{5}\).

Order of operations requires us to move from left to right, taking two fractions at a time.

\(\left( \dfrac{1}{2} ÷ \dfrac{3}{4} \right) ÷ \dfrac{3}{5} = \left( \dfrac{1}{2} \cdot \dfrac{4}{3} \right) ÷ \dfrac{3}{5} = \left( \dfrac{2}{3} \right) ÷ \dfrac{3}{5} = \dfrac{2}{3} \cdot \dfrac{5}{3} = \dfrac{10}{9}\)

Example 5.3.4

Simplify \(\dfrac{2y}{24-6y} ÷ \dfrac{y-2}{y^2 -3y-4} ÷ \dfrac{y^2+y}{3}\)

\( \begin{array} &\left( \dfrac{2y}{24-6y} ÷ \dfrac{y-2}{y^2 -3y-4} \right) ÷ \dfrac{y^2+y}{3} &= \left( \dfrac{2}{6} \cdot \dfrac{y}{\cancel{4-y}^{\textcolor{red}{-1}}} \dfrac{(\cancel{y-4})(y+1)}{y-2} \right) ÷ \dfrac{y^2+y}{3} \\ &= \left( \dfrac{1}{\cancel{3}} \cdot \dfrac{\cancel{y}}{-1} \cdot \dfrac{\cancel{y+1}}{y-2} \right) \cdot \dfrac{\cancel{3}}{\cancel{y} (\cancel{y+1})} \\ &= -\dfrac{1}{y-2} \\ &= \dfrac{1}{2-y} \end{array} \)

Try It! (Exercises)

For #1-3, divide the rational numbers without using a calculator. Give answer in reduced fraction form.

• \(\dfrac{7^{10}}{4^5} ÷ \dfrac{7^{12}}{4^6}\)
• \(\dfrac{13^8 \cdot 5^{10}}{6^7} ÷ \dfrac{13^8 \cdot 5^8}{3 \cdot 6^6 \cdot 5}\)
• \(2^9 ÷ \dfrac{4 \cdot 2^6}{8^2} ÷ \dfrac{2^3}{6}\)

For #4-20, divide and simplify.

• \(\dfrac{24u^2}{9w^8} ÷ \dfrac{12u}{45w^{10}}\)
• \(\dfrac{\frac{36a^2b^2}{25c^3}}{\frac{72ab}{5c^5}}\)
• \(\dfrac{60x^6y^{10}}{11z^5} ÷ \dfrac{9x^8}{44z^2} ÷ \dfrac{8y^7}{x}\)
• \(\dfrac{10p(p−q)^4}{3q(q−7)^2} ÷ \dfrac{2p^2(p−q)^5}{15q(q−7)^4}\)
• \(\dfrac{40v(v−2)^{12}}{33(5v−6)^9} ÷ \dfrac{24v^3(v−12)^{10}}{11(5v−6)^{10}}\)
• \(\dfrac{\frac{d−5}{20d}}{\frac{(d−5)^3}{25d^2}}\)
• \(\dfrac{\frac{4r^2−9}{r}}{2r+3}\)
• \(\dfrac{x^2+3x−18}{x^2−2x−3} ÷ \dfrac{x^2+12x+36}{x^2−6x−7}\)
• \(\dfrac{4m^3+3m^2}{8m^2} ÷ \dfrac{4m^3+7m^2+3m}{4m}\)
• \(\dfrac{\frac{12t^3}{t^2−4}}{\frac{8t^2}{4t−16}} \)
• \(\dfrac{5n+6}{\frac{5n^2−4n−12}{4−n^2}}\)
• \(\dfrac{6p^2+6p-72}{3−p} ÷ (3p + 12)\)
• \(\dfrac{y^2+2y}{4y−32} ÷ \dfrac{y+2}{12y−24} ÷ \dfrac{6y−12}{5y}\)
• \(\dfrac{w^2−25}{w^2−2w−35} ÷ \dfrac{4−4w}{w^4+2w^3} ÷ \dfrac{w^3−5w^2}{w^2−8w+7}\)
• \(\dfrac{b^4−1}{b^2−5b−6} ÷ \dfrac{b^4+b^2}{b^2−12b+36}\)
• \(\dfrac{\frac{8a^3−1}{2a^2+7a−4}}{\frac{16a}{4a+16}}\)
• \(\dfrac{\frac{64c^3+27}{16c^3}}{\frac{16c^2−9}{32c^6}}\)

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Hexingo - Operations with rational numbers Online practice for grades 7-8

Practice adding, subtracting, and multiplying a mixture of fractions and decimals (some are negative) in this fun game! To win, you need to get four correct answers in a row on the hexagon-tiled board.

You can choose to include or not include three different operations: addition, subtraction, and multiplication. The game will give you math problems that always include both a fraction and a decimal, such as 1/2 + 0.2 or −0.5 − 1/5. There are also two difficulty levels to choose from.

DIVIDING RATIONAL NUMBERS WORD PROBLEMS

Problem 1 :

A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent ?

To find how far she should travel in each descent, we have to divide 100 by 5. 

Take the reciprocal of the divisor 5.

5 ----> reciprocal ----> 1/5

Step 2 : 

Multiply 100 by 1/5

(100) x (1/5)  

Step 3 :  Simplify

(20) x (1/1)

Step 4 :  Multiply 

(20) x (1/1)  =  20    

So, she should travel 20 feet in each descent.

Problem 2 : 

A teacher wants to give pizza to her students and each student can eat 1/4 of a pizza. If there are 36 students in her class, how many pizzas does she need ?

Given :  One student can eat 1/4 of a pizza.

To find the number of pizzas required for 36 students, we have to multiply 36 by 1/4. 

36 ⋅ 1/4  =  9

So, the teacher needs 9 pizzas.

Problem 3 : 

Mason made 3/4 of a pound of trail mix. If he puts 3/8 of a pound into each bag, how many bags can Mason fill?

To find the number of bags, we have to divide 3/4 by 3/8. 

(3/4)  ÷ (3/8)  =  (3/4) ⋅ (8/3)

(3/4)  ÷ (3/8)  =  24 / 12

(3/4)  ÷ (3/8)  =  2

So, Mason fill can fill the trail mix in 2 bags. 

Problem 4 : 

Raymond bought 5 rolls of paper towels. He got 99 ⅘  meters of paper towels in all. How many meters of paper towels were on each roll ?

To find no. of meters of paper towels were on each roll, we have to divide 99 4/5 by 5. 

5 ----> reciprocal ----> 1/5

Multiply 99 4/5 by 1/5

(99 4/5) x (1/5)  =  (499/5) x (1/5)    

(99 4/5) x (1/5)  =  499/25

(99 4/5) x (1/5)  =  19 ²⁴⁄₂₅     

So, 19 ²⁴⁄₂₅  meters of paper towels were on each roll.

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Operations on Rational Numbers

Operations on rational numbers are carried out in the same way as the arithmetic operations like addition, subtraction, multiplication, and division on integers and fractions. Arithmetic operations on rational numbers with the same denominators are easy to calculate but in the case of rational numbers with different denominators, we have to operate after making the denominators the same. Rational numbers are expressed in the form of fractions, but we do not call them fractions as fractions include only positive numbers, while rational numbers include both positive and negative numbers. Fractions are a part of rational numbers, while rational numbers are a broad category that includes other types of numbers.

In this lesson, we will explore operations on rational numbers by learning about addition, subtraction, multiplication, and division of rational numbers along with their properties.

What are Operations on Rational Numbers?

Operations on rational numbers refer to the mathematical operations carrying out on two or more rational numbers . A rational number is a number that is of the form p/q, where: p and q are integers, q ≠ 0. Some examples of rational numbers are: 1/2, −3/4, 0.3 (or) 3/10, −0.7 (or) −7/10, etc.

We know about fractions and how different operators can be used on different fractions. All the rules and principles that apply to fractions can also be applied to rational numbers. The one thing that we need to remember is that rational numbers also include negatives. So, while 1/5 is a rational number, it is true that −1/5 is also a rational number. There are four basic arithmetic operations with rational numbers: addition, subtraction, multiplication, and division. Let's learn about each in detail.

Addition of Rational Numbers

Adding rational numbers can be done in the same way as adding fractions. There are two cases related to the addition of rational numbers.

• Adding rational numbers with like denominators
• Adding rational numbers with different denominators

To add two or more rational numbers with like denominators, we simply add all the numerators and write the common denominator. For example, add 1/8 and 3/8. Let us understand this with the help of a number line.

• On the number line, we start from 1/8.
• We will take 3 jumps toward the right as we are adding 3/8 to it. As a result, we reach point 4/8. 1/8 + 3/8 = (1 + 3)/8 = 4/8 =1/2
• Thus, 1/8 + 3/8 = 1/2.

When rational numbers have different denominators, the first step is to make their denominators equivalent using the LCM of the denominators. Let's consider an example. Let us add the numbers −1/3 and 3/5

• Step 1: The denominators are different in the given numbers. Let's find the LCM of 3 and 5 to find the common denominator . LCM of 3 and 5 =15
• Step 2: Find the equivalent rational number with the common denominator. To do this, multiply −1/3 with 5 and 3/5 with 3 −1/3 × 5/5 and − 5/15 = 3/5 × 3/3 = 9/15.
• Step 3: Now the denominators are the same; simply add the numerators and then copy the common denominator. Always reduce your final answer to its lowest term. −1/3+3/5=(−1/3×5/5)+(3/5×3/3) =−5/5+9/15 =4/15

Subtraction of Rational Numbers

The process of subtraction of rational numbers is the same as that of addition. While subtracting two rational numbers on a number line, we move toward the left. Let us understand this method using an example. Subtract 1/2x−1/3x

• Step 1: Find the LCM of the denominators. LCM (2, 3) = 6.
• Step 2: Convert the numbers into their equivalents with 6 as the common denominator. 1/2x × 3/3 = 3/6x = 1/3x × 2/2 = 2/6x
• Step 3: Subtract the numbers you obtained in step 2.

Multiplication of Rational Numbers

Multiplication of rational numbers is similar to how we multiply fractions. To multiply any two rational numbers, we have to follow three simple steps. Let's multiply the following rational numbers: −2/3×(−4/5). The steps to find the solution are:

• Step 1: Multiply the numerators . (−2)×(−4)=8
• Step 2: Multiply the denominators. (3)×(5)=15
• Step 3: Reduce the resulting number to its lowest term. Since it's already in its lowest term, we can leave it as is. (−23)×(−45) = (−2)×(−4)/ (3)×(5) = 8/15

Division of Rational Numbers

We have learned in the whole number division that the dividend is divided by the divisor. Dividend÷Divisor=Dividend/Divisor. While dividing any two numbers, we have to see how many parts of the divisor are there in the dividend. This is the same for the division of rational numbers as well. Let us take an example to understand it in a better way. The steps to be followed to divide two rational numbers are given below:

• Step 1: Take the reciprocal of the divisor (the second rational number). 2x/9 = 9/2x
• Step 2: Multiply it to the dividend. −4x/3 × 9/2x
• Step 3: The product of these two numbers will be the solution. (−4x × 9) / (3 × 2x) = −6

Properties of Operations on Rational Numbers

Some of the properties that apply to the operations on rational numbers are listed below:

Related Articles

Check out a few more interesting articles related to the operations of rational numbers.

• Decimal Representation of Irrational Numbers
• Irrational Numbers
• Rationalize the Denominator
• Is pi a rational or Irrational Number

Important Notes

• Identity property does not hold true for subtraction and division of rational numbers.
• Closure property holds true for all four operations of rational numbers.
• Commutative property and associative property holds true for the addition and multiplication of rational numbers.
• Inverse property does not hold true for subtraction and division of rational numbers.x/y−(−x/y)≠0, x/y:y/x≠1

Examples of Operations of Rational Numbers

Example 1: Using the properties of rational numbers, determine the difference between −5/7 and 3/7.

The given rational numbers have a common denominator. Thus, we will subtract the numerators and retain the same denominator.

= −5/7−3/7 = (−5−3)/7 = −8/7

Therefore, the difference is −8/7.

Example 2: Saira uses 3/5 of the flour if she has to bake a full cake. How much flour will she use to bake 1/6 portion of the cake?

Total flour to bake a full cake = 3/5

Using operations on rational numbers,

Amount of flour used to bake1/6 portion of the cake = 3/5×1/6=(3×1)/(5×6)=3/30=1/10

Therefore, Saira would have to use 1/10 of the flour.

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Practice Questions on Operations of Rational Numbers

Faqs on operations of rational numbers, what are the effects of different operations on rational and irrational numbers.

• The resultant of the addition of a rational number and an irrational number is an irrational number only as it doesn't affect the non-recurring and non-terminating nature of the irrationals.
• The sum of two rational numbers is a rational number.
• The sum of a rational number and an irrational number is irrational.
• The sum of two irrational numbers is an irrational number.
• The product of two rational numbers is a rational number.
• The product of a rational number and an irrational number is an irrational number.
• The product of two irrational numbers is an irrational number.

How Do You Perform Operations on Rational Numbers?

We perform the operations on rational numbers as follows:

• Addition and subtraction of rational numbers: In case, the denominators are the same, just add or subtract directly. In case, the denominators are different, take LCM to make the denominators the same and then solve.
• Multiplication of rational numbers: Multiply the numerators and multiply the denominators. Reduce the fraction so obtained in its lowest form.
• Division of Rational Numbers: Multiply the reciprocal of the divisor with the dividend.

What Are the Properties of Addition of Rational Numbers?

Properties of addition of rational numbers are described below:

• Addition of two rational numbers is also a rational number. (closure property)
• Three rational numbers can be added in any order. (Associative property)
• Two rational numbers can be rearranged internally without affecting the addition of numbers. (Commutative property)
• 0 is the additive identity of any rational number.
• Additive inverse of a rational number in the form of p/q is −pq.

Does the Identity Property Hold True for the Subtraction of Rational Numbers?

Identity property partially holds true in the case of subtraction of rational numbers, as x/y − 0 = x/y, but 0 − x/y ≠ x/y.

What Is the Rule for Subtracting Rational Numbers?

To subtract any two rational numbers

• Step 1: Check if the denominators are the same.
• Step 2: Make the denominators the same by taking the LCM of the denominators.
• Step 3: Subtract the given numbers by subtracting their numerators, leaving the denominator the same.

What Is the Difference Between Operations on Fractions and Operations on Rational Numbers?

In operations on rational numbers, we need to use the rules of operations on integers as well as operations on fractions, because rational numbers include negative numbers also. For positive rational numbers, the process of applying operations is the same as that of fractions.

Does the Inverse Property Hold True for the Division of Rational Numbers?

No, the inverse property does not hold true for the division of rational numbers, that's why we call it multiplicative inverse and not division inverse. Because if we divide x/y by y/x, we won't get 1 as the answer. Let's check. x/y ÷ y/x = x/y × x/y = x 2 /y 2 ≠16.

How To Add Two Negative Rational Numbers?

Let us take an example to understand how to add two negative rational numbers. Add: −1/2+(−3/4)

• Whenever there is a positive sign outside the bracket, we consider the sign of individual terms inside the bracket. So here, we can write it as −1/2 − 3/4.
• Now take the LCM of the denominators to make these terms like. LCM (2,4)=4
• Solve the numerators and write the final answer. −2/4 − 3/4 = −5/4

This is how we add two negative rational numbers.

How To Subtract Two Negative Rational Numbers?

Let us take an example to understand how to subtract two negative rational numbers. Subtract −3/7−(−4/3)

• Whenever there is a negative sign outside the bracket, we change the sign of individual terms inside the bracket. So here, we write −4/3 as +4/3.
• Now take the LCM of the denominators to make these terms like. LCM(7,3)=21
• −12/21 + 28/21. Solve the numerators and write the final answer. −12/21 + 28/21 = 16/21

This is how we subtract two negative rational numbers.

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Adding and subtracting rational numbers to solve problems.

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• Grade Levels 7th Grade
• Related Academic Standards CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.1.7.E.1 Apply and extend previous understandings of operations with fractions to operations with rational numbers.
• Assessment Anchors M07.A-N.1 Apply and extend previous understandings of operations to add, subtract, multiply, and divide rational numbers. M07.B-E.2 Solve real-world and mathematical problems using numerical and algebraic expressions, equations, and inequalities.
• Eligible Content M07.A-N.1.1.1 Apply properties of operations to add and subtract rational numbers, including real-world contexts. M07.B-E.2.1.1 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50 an hour (or 1.1 × $25 = $27.50).
• Competencies

Students will compute and solve problems using rational numbers. They will:

• add and subtract rational numbers.
• solve real-world problems by adding and subtracting rational numbers.

Essential Questions

• How is mathematics used to quantify, compare, represent, and model numbers? 
• How are relationships represented mathematically?
• How can expressions, equations and inequalities, be used to quantify, solve, and/or analyze mathematical situations?
• What makes a tool and/or strategy appropriate for a given task?
• How can recognizing repetition or regularity assist in solving problems more efficiently?
• Rational Number: A number expressible in the form a / b, where a and b are integers, and b ≠ 0.
• Repeating Decimal: The decimal form of a rational number in which the decimal digits repeat in an infinite pattern.

60–90 minutes

Prerequisite Skills

• Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx )
• Lesson 2 Small-Group Practice worksheet ( M-7-5-2_Small Group Practice and KEY.docx )
• Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx )
• Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx )
• Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx )

Related Unit and Lesson Plans

• Computing and Problem Solving with Rational Numbers
• Adding and Subtracting Rational Numbers on a Number Line
• Multiplying and Dividing Rational Numbers to Solve Problems

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

• http://ca.ixl.com/math/grade-7/add-and-subtract-rational-numbers

IXL’s Grade 7 Add and Subtract Rational Numbers will give students additional practice with addition and subtraction of rational numbers.

• http://ca.ixl.com/math/grade-8/add-and-subtract-rational-numbers-word-problems

IXL’s Grade 8 Add and Subtract Rational Numbers: Word Problems will give students additional practice with solving word problems that involve rational numbers.

Formative Assessment

• The modeling activity can be used to assess students’ prior knowledge and understanding regarding addition of rational numbers with unlike denominators.
• Activity 1 can be used to assess each student’s ability to create a word problem involving the addition and/or subtraction of rational numbers while also understanding the solution process.
• Use the exit ticket to quickly evaluate student mastery.

Suggested Instructional Supports

Instructional procedures.

As students come into class, have them evaluate the following expressions using a number line.

• 0.75 + 2.95      (3.7)

Walk around the classroom as students are working through the example problems. Briefly discuss the answers and make sure students are comfortable modeling addition and subtraction of rational numbers on a number line before moving on.

“In Lesson 1 of this unit, we learned how to model addition and subtraction of rational numbers on a number line. Today, we are going to focus on performing these computations without the use of a number line. We will then use these skills to solve some real-life problems.”

Computations: Adding and Subtracting Rational Numbers

Before presenting some real-world problems, give students the opportunity to practice adding and subtracting rational numbers without the help of a number line. If necessary, go over the following examples together as a class.

the other is not. Often, when computing with fractions, it is best to write all numbers in fraction form.”

common denominator. The lowest common denominator in this case would be 5.”

numerators as indicated. The denominator will stay as is.”

it is, but we may want to rewrite the fraction as a mixed number to get a better idea of the value.”

Example 2:      − 4.64 + 9.85

• −4.64 + 9.85     “Think about the number line. Based on the signs of each addend, do

you suspect our final answer here will be positive or negative?” (Positive, the absolute value of 9.85 is larger than the absolute value of −4.64.)

• Think:            9.85 – 4.64    
•                             

Distribute the Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx ). Instruct students to complete the worksheet individually. Walk around the room as students work to be sure they are on task and performing the computations accurately. Following the worksheet, provide time for students to discuss any problems they encountered, questions they have, or revelations they discovered. First, ask students to describe the computation process used to find each sum or difference. Then confirm their understanding by restating the correct process.

Problem Solving with Rational Numbers

Now it is time for students to apply their understanding of computation to solving real-world problems. Discuss the following examples together as a class.

• “The amount by which his savings have decreased is equal to the difference of 1018.20 and 920.45, written as 1018.20 – 920.45 or 97.75. Thus, Steven’s savings decreased by $97.75.”

Distribute Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx ). Have students discuss the solution process for each example problem in a manner similar to the process demonstrated above. Confirm the correct ideas students express. Then say: “Look through the problems you just received. Think of how the example word problems can be solved. Do you need to add or subtract the rational numbers? How will you go about doing this for fractions with unlike denominators, or for mixed numbers?”

Activity 1: Write-Pair-Share

Ask the whole class to think of some real-world contexts that involve the addition or subtraction of rational numbers. Students should make a list of at least five real-world contexts and provide one word problem. Ask students to share their ideas with a partner. Give students about 5 minutes to share contexts and word problems. During this time, each partner may ask questions of the other partner. Then, the whole class can reconvene. One member from each partner group will share the list of real-world contexts and word problems with the class. The teacher may wish to post the real-world contexts and word problems in a file on the class Web page or use them as a classroom display. These student examples would then serve as a reference tool.

Have students complete Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx ) at the close of the lesson to evaluate students’ level of understanding.

Use the suggestions in the Routine section to review lesson concepts throughout the school year. Use the small-group suggestions for any students who might benefit from additional instruction. Use the Expansion section to challenge students who are ready to move beyond the requirements of the standard.

• Routine: Throughout the school year, encourage students to be on the lookout for real-world situations that involve the addition or subtraction of rational numbers. Students can present the problems to the teacher, who will facilitate class participation in solving the rational number problem.
• Small Groups: Students who need additional practice can be pulled into small groups to work on the Lesson 2 Small-Group Practice worksheet ( M-7-5-3_Small Group Practice and KEY.docx ). Students can work on the matching together or work individually and compare answers when done.
• Expansion: Students who are prepared for a greater challenge can be given the Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx ). The worksheet includes more difficult numeric expressions involving rational numbers.

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