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Study Guides > Intermediate Algebra

Read: Multiplying Fractions

Learning Objectives

  • Multiply two or more fractions
  • Multiply a fraction by a whole number

Multiply Fractions

Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions.   There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions. When you multiply a fraction by a fraction, you are finding a “fraction of a fraction.” Suppose you have 34\Large\frac{3}{4} of a candy bar and you want to find 12\Large\frac{1}{2} of the 34\Large\frac{3}{4}: 3 out of four boxes are shaded. This is 3/4. By dividing each fourth in half, you can divide the candy bar into eighths. Six of 8 boxes are shaded. This is 6/8. Then, choose half of those to get 38\Large\frac{3}{8}. Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3/8. In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.

Multiplying Two Fractions

ab⋅cd=a⋅cb⋅d=product of the numeratorsproduct of the denominators\Large\frac{a}{b}\cdot\Large\frac{c}{d}=\Large\frac{a\cdot c}{b\cdot d}=\Large\frac{\text{product of the numerators}}{\text{product of the denominators}}

Multiplying More Than Two Fractions

ab⋅cd⋅ef=a⋅c⋅eb⋅d⋅f\Large\frac{a}{b}\cdot\Large\frac{c}{d}\cdot\Large\frac{e}{f}=\Large\frac{a\cdot c\cdot e}{b\cdot d\cdot f}

Example

Multiply 23⋅45\Large\frac{2}{3}\cdot\Large\frac{4}{5}

Answer: Multiply the numerators and multiply the denominators.

2⋅43⋅5\Large\frac{2\cdot 4}{3\cdot 5}

Simplify, if possible. This fraction is already in lowest terms.

815\Large\frac{8}{15}

Answer

815\Large\frac{8}{15}

To review: if a fraction has common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors. For example,
  • Given 815\Large\frac{8}{15}, the factors of 88 are: 1,2,4,81, 2, 4, 8 and the factors of 1515 are: 1,3,5,151, 3, 5, 15.  815\Large\frac{8}{15} is simplified because there are no common factors of 88 and 1515.
  • Given 1015\Large\frac{10}{15}, the factors of 1010 are: 1,2,5,101, 2, 5, 10 and the factors of 1515 are: 1,3,5,151, 3, 5, 15. 1015\Large\frac{10}{15} is not simplified because 55 is a common factor of 1010 and 1515.
You can simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply. In the following video you will see an example of how to multiply two fractions, then simplify the answer. https://youtu.be/f_L-EFC8Z7c

Think About It

Multiply 23⋅14⋅35\Large\frac{2}{3}\cdot\Large\frac{1}{4}\cdot\Large\frac{3}{5}. Simplify the answer. What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together. [practice-area rows="2"][/practice-area]

Answer: Multiply the numerators and multiply the denominators.

2⋅1⋅33⋅4⋅5\Large\frac{2\cdot 1\cdot 3}{3\cdot 4\cdot 5}

Simplify first by canceling (dividing) the common factors of 33 and 22. 33 divided by 33 is 11, and 22 divided by  22 is 11.

2⋅1⋅33⋅(2⋅2)⋅52⋅1⋅33⋅(2⋅2)⋅5110\begin{array}{c}\Large\frac{2\cdot 1\cdot3}{3\cdot (2\cdot 2)\cdot 5}\\\Large\frac{\cancel{2}\cdot 1\cdot\cancel{3}}{\cancel{3}\cdot (\cancel{2}\cdot 2)\cdot 5}\\\Large\frac{1}{10}\end{array}

Answer

23⋅14⋅35[/latex]=[latex]110\Large\frac{2}{3}\cdot\Large\frac{1}{4}\cdot\Large\frac{3}{5}[/latex] = [latex]\Large\frac{1}{10}

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