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Doing the Manipulative Mathematics activity “Model Fraction Multiplication” will help you develop a better understanding of multiplying fractions.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with 3 4 .

A rectangle made up of four squares in a row. The first three squares are shaded.

Now we’ll take 1 2 of 3 4 .

A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.

Notice that now, the whole is divided into 8 equal parts. So 1 2 · 3 4 = 3 8 .

To multiply fractions, we multiply the numerators and multiply the denominators.

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.

When multiplying fractions , the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In [link] , we will multiply negative and a positive, so the product will be negative.

Multiply: 11 12 · 5 7 .

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

11 12 · 5 7 Determine the sign of the product; multiply. 11 · 5 12 · 7 Are there any common factors in the numerator and the denominator? No 55 84

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Multiply: 10 28 · 8 15 .

4 21

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Multiply: 9 20 · 5 12 .

3 16

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When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a , can be written as a 1 . So, for example, 3 = 3 1 .

Multiply: 12 5 ( −20 x ) .

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

12 5 ( −20 x )
Write 20 x as a fraction. 12 5 ( 20 x 1 )
Multiply.
Rewrite 20 to show the common factor 5 and divide it out. .
Simplify. 48 x
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Multiply: 11 3 ( −9 a ) .

−33 a

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Multiply: 13 7 ( −14 b ) .

−36 b

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Divide fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal    of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of 2 3 is 3 2 .

Notice that 2 3 · 3 2 = 1 . A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of 10 7 is 7 10 , since 10 7 ( 7 10 ) = 1 .

Reciprocal

The reciprocal of a b is b a .

A number and its reciprocal multiply to one a b · b a = 1 .

Doing the Manipulative Mathematics activity “Model Fraction Division” will help you develop a better understanding of dividing fractions.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

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, we multiply the first fraction by the reciprocal of the second.

We need to say b 0 , c 0 and d 0 to be sure we don’t divide by zero!

Divide: 2 3 ÷ n 5 .

Solution

2 3 ÷ n 5 To divide, multiply the first fraction by the reciprocal of the second. 2 3 · 5 n Multiply. 10 3 n

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Divide: 3 5 ÷ p 7 .

21 5 p

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Divide: 5 8 ÷ q 3 .

15 8 q

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Find the quotient: 7 8 ÷ ( 14 27 ) .

Solution

7 18 ÷ ( 14 27 )
To divide, multiply the first fraction by the reciprocal of the second. 7 18 27 14
Determine the sign of the product, and then multiply.. 7 27 18 14
Rewrite showing common factors. .
Remove common factors. 3 2 2
Simplify. 3 4
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Questions & Answers

At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
Andrew Reply
divide 3x⁴-4x³-3x-1 by x-3
Ritik Reply
how to multiply the monomial
Ceny Reply
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
Seera Reply
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Juned Reply
Brandon has a cup of quarters and dimes with a total of 5.55$. The number of quarters is five less than three times the number of dimes
ashley Reply
app is wrong how can 350 be divisible by 3.
Raheem Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna Reply
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
Lorris Reply
I'm getting "math processing error" on math problems. Anyone know why?
Ray Reply
Can you all help me I don't get any of this
Jade Reply
4^×=9
Alberto Reply
Did anyone else have trouble getting in quiz link for linear inequalities?
Sireka Reply
operation of trinomial
Justin Reply
y=2×+9
Jacob Reply
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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