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Multiplying and Dividing Fractions

Author: Sophia

what's covered
This tutorial covers multiplying and dividing fractions, through the exploration of:

Table of Contents

1. Review of PEMDAS, the Order of Operations

As you may recall, PEMDAS is the acronym that is used to remember the order of operations. PEMDAS stands for:

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

The order of operations applies to all numbers, including fractions.


2. Multiplying and Dividing Fractions

Multiplication and division involving fractions is useful when converting between measurements, which is common in many scientific fields, such as chemistry, physics, and nursing.

key concept
When you multiply fractions, you multiply the numerators together to find the numerator of your answer. Similarly, you multiply the denominators together to find the denominator of your answer.

EXAMPLE

Suppose you want to multiply 3/4 times 1/5.

3 over 4 cross times 1 fifth Our expression
3 over 4 cross times 1 fifth equals 3 over space First, multiply the numerators to find our answer's numerator. 3 times 1 equals 3.
3 over 4 cross times 1 fifth equals 3 over 20 Next, multiply the denominators to find our answer's denominator. 4 times 5 equals 20.
3 over 20 Our Solution

Dividing fractions, on the other hand, is equivalent to multiplying by the reciprocal. Finding the reciprocal of a fraction means flipping it, or switching the numerator and the denominator.

EXAMPLE

Suppose you want to divide 3/8 by 1/2.

3 over 8 divided by 1 half Our Expression
3 over 8 cross times 2 over 1 First, find the reciprocal of 1 half by flipping it. This is 2 over 1. Change the sign to multiplication.
3 over 8 cross times 2 over 1 equals 6 over space Next, multiply the numerators. 3 times 2 is 6.
3 over 8 cross times 2 over 1 equals 6 over 8 Next, multiply the denominators. 8 times 1 is 8.
6 over 8 Our Solution

hint
It is important to note that you flipped your second fraction, and you changed the division sign to multiplication.

big idea
You need to make sure that the fraction is simplified, which means that you need to cancel out any common factors of both the numerator (6) and the denominator (8).

term to know
Reciprocal (of a Fraction)
A fraction in which the numerator and denominator have been switched


3. Simplifying Fractions

A simplified fraction is a fraction in which the numerator and the denominator have no common factors other than 1.

did you know
You always want to write fractions in their simplest form so they are easier to compare and calculate with.

EXAMPLE

50/100 can be simplified to 1/2, because both the numerator and the denominator are divisible by 50. It’s much easier to say that you ate half of your dinner rather than fifty-hundredths of your dinner!

Referring back to the previous example in which you arrived at an answer of 6/8, how can you simplify this fraction?

6 over 8 Our Expression
6 over 8 equals fraction numerator 2 cross times 3 over denominator 2 cross times 2 cross times 2 end fraction First, see if the numerator and denominator have any common factors. Expand both numbers into their prime factors. In this case, the prime factors of 6 are 3 and 2. The prime factors of 8 are 2, 2, and 2.
6 over 8 equals fraction numerator up diagonal strike 2 cross times 3 over denominator up diagonal strike 2 cross times 2 cross times 2 end fraction Since there is at least one 2 in both the numerator and the denominator, you can cancel them out.
6 over 8 equals fraction numerator 3 over denominator 2 cross times 2 end fraction This leaves 3 in the numerator, and 2 times 2 in the denominator.
6 over 8 equals 3 over 4 Evaluate 2 times 2, which is 4.
3 over 4 Our Solution.

3 over 4 is the simplest form of 6 over 8.


4. Using the Order of Operations with Fractions

You can also use the order of operations, or PEMDAS, with fractions.

EXAMPLE

Suppose you want to simplify the following expression: 2 over 9 divided by open parentheses 1 third close parentheses squared cross times 1 fourth.

2 over 9 divided by open parentheses 1 third close parentheses squared cross times 1 fourth Our Expression
2 over 9 divided by stack open parentheses 1 third close parentheses squared with underbrace below cross times 1 fourth First, evaluate the exponent. 1/3 times 1/3 is 1/9
stack 2 over 9 divided by 1 over 9 with underbrace below cross times 1 fourth Next, multiply and divide from left to right. When you divide 2/9 by 1/9, you need to multiply by the reciprocal of the second fraction, which is 9/1
2 over 9 cross times 9 over 1 cross times 1 fourth equals 18 over space Since we just have multiplication remaining, we can multiply all 3 numerators together. 2 times 9 times 1 is 18.
2 over 9 cross times 9 over 1 cross times 1 fourth equals 18 over 36 Then, we can multiply all three denominators together. 9 times 1 times 4 equals 36
18 over 36 equals fraction numerator 2 cross times 3 cross times 3 over denominator 2 cross times 2 cross times 3 cross times 3 end fraction Finally, we want to simplify the fraction by finding the prime factors of the numerator and denominator.
18 over 36 equals fraction numerator up diagonal strike 2 cross times up diagonal strike 3 cross times up diagonal strike 3 over denominator 2 cross times up diagonal strike 2 cross times up diagonal strike 3 cross times up diagonal strike 3 end fraction One 2 and two 3's cancel out of both the numerator and denominator
1 half Our solution

summary
Today you learned that you can use the order of operations, or PEMDAS, with all numbers, including fractions. You learned how to multiply fractions by multiplying the numerators and the denominators together straight across to find the numerator and denominator of your answer. You also learned that when dividing fractions, you keep the first fraction the same, change the division sign to multiplication, and then find the reciprocal of the second fraction by flipping it. Lastly, you learned that to simplify fractions, you can cancel out any common factors of both the numerator and denominator.

Source: This work is adapted from Sophia author Colleen Atakpu.

Terms to Know
Reciprocal (of a Fraction)

A fraction in which the numerator and denominator have been switched.