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Multiplying and Dividing Fractions

Multiplying and Dividing Fractions

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In this lesson, students will learn how to multiply and divide fractions.

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Tutorial
This tutorial covers multiplying and dividing fractions, through the exploration of:
  1. Review of PEMDAS, the Order of Operations
  2. Multiplying and Dividing Fractions
  3. Simplifying Fractions
  4. Using the Order of Operations with Fractions


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.

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.

Suppose you want to multiply 3/4 times 1/5. In this problem, the 3 and the 1 are the numerators of the fractions, and the 4 and the 5 are the denominators. First, you multiply the numerators together to find the numerator of your answer: 3 times 1 equals 3. Next, you multiply the denominators together to find the denominator of your answer: 4 times 5 equals 20. Your final answer is 3/20.
3 over 4 cross times 1 fifth equals 3 over 20
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.

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

Suppose you want to divide 3/8 by 1/2. First, you need to find the reciprocal of 1/2, so by flipping it, your division problem becomes 3/8 times 2/1. Now you can multiply your fractions in the same way that you did in the first example. You multiply your numerators together (3 times 2 equals 6) and then you multiply your denominators together (8 times 1 equals 8). The resulting answer is 6/8.
table attributes columnalign left end attributes row cell 3 over 8 divided by 1 half end cell row cell 3 over 8 cross times 2 over 1 equals 6 over 8 end cell end table

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

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).


3. Simplifying Fractions

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

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

For 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?

First, you need to see if the numerator 6 and the denominator 8 have any common factors. To do this you can expand both numbers into their prime factors—the prime numbers that can be multiplied together to result in the original number. In this case, the prime factors of 6 are 2 and 3, and the prime factors of 8 are 2, 2, and 2.

6 over 8 equals fraction numerator 2 cross times 3 over denominator 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, leaving you with a 3 in the numerator and 2 times 2 in the denominator, which equals the simplified fraction 3/4.

6 over 8 equals fraction numerator 3 over denominator 2 cross times 2 end fraction equals 3 over 4


4. Using the Order of Operations with Fractions

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

Suppose you want to simplify the following expression:
2 over 9 divided by left parenthesis 1 third right parenthesis squared cross times 1 fourth
Step 1: You start with your exponent operation and calculate 1/3 squared, or 1/3 times 1/3. Multiplying the numerators and the denominators together equals 1/9. Therefore, your expression becomes:
2 over 9 divided by 1 over 9 cross times 1 fourth
Step 2: Next, you perform multiplication and division from left to right in the order that they appear, beginning with dividing 2/9 by 1/9. Remember that when you divide fractions, it’s the same as multiplying by the reciprocal. Therefore, the expression becomes:
2 over 9 cross times 9 over 1 cross times 1 fourth
Step 3: Since you only have multiplication remaining, you can multiply all three numerators together—2 times 9 times 1, which equals 18—and all three denominators together—9 times 1 times 4, which equals 36.
2 over 9 cross times 9 over 1 cross times 1 fourth equals 18 over 36
Step 4: Lastly, you can simplify 18/36 by writing the numerator and denominator as prime factors. The prime factors of 18 are 2, 3, and 3, and the prime factors of 36 are 2, 2, 3, and 3. You can cross out one 2 and two 3s in both the numerator and the denominator. Since you have no more factors remaining in the numerator, this means your numerator is 1, and since your denominator has one 2 remaining, your final answer is 1/2.
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 equals 1 half
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.