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The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division . The quotient of a and b is the result we get from dividing a by b , or a b .

Translate the English phrase into an algebraic expression: the quotient of the difference of m and n , and p .

Solution

We are looking for the quotient of the difference of m and n , and p . This means we want to divide the difference of m and n by p .

m n p
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Translate the English phrase into an algebraic expression: the quotient of the difference of a and b , and cd .

a b c d

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Translate the English phrase into an algebraic expression: the quotient of the sum of p and q , and r

p + q r

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Key concepts

  • Equivalent Fractions Property: If a , b , c are numbers where b 0 , c 0 , then
    a b = a · c b · c and a · c b · c = a b .
  • Fraction Division: If a , b , c and d are numbers where b 0 , c 0 , and d 0 , then a b ÷ c d = a b · d c . To divide fractions, multiply the first fraction by the reciprocal of the second.
  • Fraction Multiplication: If a , b , c and d are numbers where b 0 , and d 0 , then a b · c d = a c b d . To multiply fractions, multiply the numerators and multiply the denominators.
  • Placement of Negative Sign in a Fraction: For any positive numbers a and b , a b = a b = a b .
  • Property of One: a a = 1 ; Any number, except zero, divided by itself is one.
  • Simplify a Fraction
    1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers first.
    2. Simplify using the equivalent fractions property by dividing out common factors.
    3. Multiply any remaining factors.
  • Simplify an Expression with a Fraction Bar
    1. Simplify the expression in the numerator. Simplify the expression in the denominator.
    2. Simplify the fraction.

Practice makes perfect

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

3 8

6 16 , 9 24 , 12 32 answers may vary

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5 9

10 18 , 15 27 , 20 36 answers may vary

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Simplify Fractions

In the following exercises, simplify.

Multiply Fractions

In the following exercises, multiply.

( 14 15 ) ( 9 20 )

21 50

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( 63 84 ) ( 44 90 )

11 30

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( 63 60 ) ( 40 88 )

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Divide Fractions

In the following exercises, divide.

7 9 ÷ ( 7 4 )

1

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5 18 ÷ ( 15 24 )

4 9

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8 u 15 ÷ 12 v 25

10 u 9 v

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3 4 ÷ ( −12 )

1 16

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In the following exercises, simplify.

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

8 · 3 + 2 · 9 14 + 3

2 8 17

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9 · 6 4 · 7 22 + 3

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5 · 6 3 · 4 4 · 5 2 · 3

3 5

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8 · 9 7 · 6 5 · 6 9 · 2

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7 · 4 2 ( 8 5 ) 9 · 3 3 · 5

11 6

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9 · 7 3 ( 12 8 ) 8 · 7 6 · 6

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9 ( 8 2 ) 3 ( 15 7 ) 6 ( 7 1 ) 3 ( 17 9 )

5 2

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8 ( 9 2 ) 4 ( 14 9 ) 7 ( 8 3 ) 3 ( 16 9 )

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Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

the quotient of r and the sum of s and 10

r s + 10

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the quotient of A and the difference of 3 and B

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the quotient of the difference of x and y , and 3

x y −8

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the quotient of the sum of m and n , and 4 q

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Everyday math

Baking. A recipe for chocolate chip cookies calls for 3 4 cup brown sugar. Imelda wants to double the recipe. How much brown sugar will Imelda need? Show your calculation. Measuring cups usually come in sets of 1 4 , 1 3 , 1 2 , and 1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.

1 1 2 cups answers will vary

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Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs 2 3 cup of condensed milk. How much condensed milk will Nina need? Show your calculation. Measuring cups usually come in sets of 1 4 , 1 3 , 1 2 , and 1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for 4 pans of fudge.

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Portions Don purchased a bulk package of candy that weighs 5 pounds. He wants to sell the candy in little bags that hold 1 4 pound. How many little bags of candy can he fill from the bulk package?

20 bags

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Portions Kristen has 3 4 yards of ribbon that she wants to cut into 6 equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?

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Writing exercises

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.

Answers may vary

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Give an example from everyday life that demonstrates how 1 2 · 2 3 is 1 3 .

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Explain how you find the reciprocal of a fraction.

Answers may vary

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Explain how you find the reciprocal of a negative number.

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Self check

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

A table is shown that is made up of four columns and seven rows. The first row reads “I can…” in the first column, “Confidently” in the second column, “With some help” in the third column and “No – I don’t get it” in the last column. The next row down in the first column reads “find equivalent fractions”, under this reads “simplify fractions”, under this reads “multiply fractions”, under this reads “divide fractions”, under this reads “Simplify expressions written with a fraction bar” and under this reads “translate phrases to expressions with fractions.”

After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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