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Study Guides > Prealgebra

Multiplying Fractions

Learning Outcomes

  • Use a diagram to model multiplication of positive and negative fractions
  • Multiply fractions and integer expressions that contain variables

A model may help you understand multiplication of fractions. We will use fraction tiles to model [latex]\frac{1}{2}\cdot \frac{3}{4}[/latex]. To multiply [latex]\frac{1}{2}[/latex] and [latex]\frac{3}{4}[/latex], think [latex]\frac{1}{2}[/latex] of [latex]\frac{3}{4}[/latex]. Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three [latex]\frac{1}{4}[/latex] tiles evenly into two parts, we exchange them for smaller tiles.

A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths. We see [latex]\frac{6}{8}[/latex] is equivalent to [latex]\frac{3}{4}[/latex]. Taking half of the six [latex]\frac{1}{8}[/latex] tiles gives us three [latex]\frac{1}{8}[/latex] tiles, which is [latex]\frac{3}{8}[/latex]. Therefore, [latex-display]\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}[/latex-display] Doing the Manipulative Mathematics activity "Model Fraction Multiplication" will help you develop a better understanding of how to multiply fractions.

Example

Use a diagram to model [latex]\frac{1}{2}\cdot \frac{3}{4}[/latex] Solution: First shade in [latex]\frac{3}{4}[/latex] of the rectangle. A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. We will take [latex]\frac{1}{2}[/latex] of this [latex]\frac{3}{4}[/latex], so we heavily shade [latex]\frac{1}{2}[/latex] of the shaded region. A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded. Notice that [latex]3[/latex] out of the [latex]8[/latex] pieces are heavily shaded. This means that [latex]\frac{3}{8}[/latex] of the rectangle is heavily shaded. Therefore, [latex]\frac{1}{2}[/latex] of [latex]\frac{3}{4}[/latex] is [latex]\frac{3}{8}[/latex], or [latex]\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}[/latex].

Try it

**Don't remove this one. ** Use a diagram to model: [latex]\frac{1}{2}\cdot \frac{3}{5}[/latex]

Answer: [latex-display]\frac{3}{10}[/latex-display]

#146020 [ohm_question height="270"]146020[/ohm_question]
Look at the result we got from the model in the example above. We found that [latex]\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}[/latex]. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?
[latex]\frac{1}{2}\cdot \frac{3}{4}[/latex]
Multiply the numerators, and multiply the denominators. [latex]\frac{1}{2}\cdot \frac{3}{4}[/latex]
Simplify. [latex]\frac{3}{8}[/latex]
This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Fraction Multiplication

If [latex]a,b,c,\text{ and }d[/latex] are numbers where [latex]b\ne 0\text{ and }d\ne 0[/latex], then [latex-display]\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}[/latex-display]

Example

Multiply, and write the answer in simplified form: [latex]\frac{3}{4}\cdot \frac{1}{5}[/latex]

Answer: Solution:

[latex]\frac{3}{4}\cdot \frac{1}{5}[/latex]
Multiply the numerators; multiply the denominators. [latex]\frac{3\cdot 1}{4\cdot 5}[/latex]
Simplify. [latex]\frac{3}{20}[/latex]
There are no common factors, so the fraction is simplified.

Try It

#146021 [ohm_question height="270"]146021[/ohm_question]
When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In the next example, we will multiply two negatives, so the product will be positive.

Example

Multiply, and write the answer in simplified form: [latex]-\frac{5}{8}\left(-\frac{2}{3}\right)[/latex]

Answer: Solution:

[latex]-\frac{5}{8}\left(-\frac{2}{3}\right)[/latex]
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators. [latex]\frac{5\cdot 2}{8\cdot 3}[/latex]
Simplify. [latex]\frac{10}{24}[/latex]
Look for common factors in the numerator and denominator. Rewrite showing common factors. [latex]\frac{5\cdot\color{red}{2}}{12\cdot\color{red}{2}}[/latex]
Remove common factors. [latex]\frac{5}{12}[/latex]
Another way to find this product involves removing common factors earlier.
[latex]-\frac{5}{8}\left(-\frac{2}{3}\right)[/latex]
Determine the sign of the product. Multiply. [latex]\frac{5\cdot 2}{8\cdot 3}[/latex]
Show common factors and then remove them. [latex]\frac{5\cdot\color{red}{2}}{4\cdot\color{red}{2}\cdot3}[/latex]
Multiply remaining factors. [latex]\frac{5}{12}[/latex]
We get the same result.

Try it

#146022 [ohm_question height="270"]146022[/ohm_question]
The following video provides more examples of how to multiply fractions, and simplify the result. https://youtu.be/f_L-EFC8Z7c

Example

Multiply, and write the answer in simplified form: [latex]-\frac{14}{15}\cdot \frac{20}{21}[/latex]

Answer: Solution:

[latex]-\frac{14}{15}\cdot \frac{20}{21}[/latex]
Determine the sign of the product; multiply. [latex]-\frac{14}{15}\cdot \frac{20}{21}[/latex]
Are there any common factors in the numerator and the denominator? We know that [latex]7[/latex] is a factor of [latex]14[/latex] and [latex]21[/latex], and [latex]5[/latex] is a factor of [latex]20[/latex] and [latex]15[/latex].
Rewrite showing common factors. [latex]--\frac{2\cdot\color{red}{7}\cdot4\cdot\color{blue}{5}}{3\cdot\color{blue}{5}\cdot3\cdot\color{red}{7}}[/latex]
Remove the common factors. [latex]-\frac{2\cdot 4}{3\cdot 3}[/latex]
Multiply the remaining factors. [latex]-\frac{8}{9}[/latex]

 

Try it

#146023 [ohm_question height="270"]146023[/ohm_question]
The following video shows another example of multiplying fractions that are negative. https://youtu.be/yUdJ46pTblo When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, [latex]a[/latex], can be written as [latex]\frac{a}{1}[/latex]. So, [latex]3=\frac{3}{1}[/latex], for example.

example

Multiply, and write the answer in simplified form:
  1. [latex]\frac{1}{7}\cdot 56[/latex]
  2. [latex]\frac{12}{5}\left(-20x\right)[/latex]

Answer: Solution:

1.
[latex]\frac{1}{7}\cdot 56[/latex]
Write [latex]56[/latex] as a fraction. [latex]\frac{1}{7}\cdot \frac{56}{1}[/latex]
Determine the sign of the product; multiply. [latex]\frac{56}{7}[/latex]
Simplify. [latex]8[/latex]
2.
[latex]\frac{12}{5}\left(-20x\right)[/latex]
Write [latex]−20x[/latex] as a fraction. [latex]\frac{12}{5}\left(\frac{-20x}{1}\right)[/latex]
Determine the sign of the product; multiply. [latex]-\frac{12\cdot 20\cdot x}{5\cdot 1}[/latex]
Show common factors and then remove them. [latex]--\frac{12\cdot4\color{red}{\cdot 5x}}{\color{red}{5}\cdot1}[/latex]
Multiply remaining factors; simplify. [latex]−48x[/latex]

Try it

#146024 [ohm_question height="270"]146024[/ohm_question] #146025 [ohm_question height="270"]146025[/ohm_question]
Watch teh following video to see more examples of how to multiply a fraction and a whole numebr, https://youtu.be/Rxz7OUzNyV0

Licenses & Attributions

CC licensed content, Original

  • Ex 2: Multiply Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Question ID: 146020, 146021, 146022, 146023, 146024, 146025. Provided by: # Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.

CC licensed content, Shared previously

  • Ex 1: Multiply Fractions (Basic). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
  • Ex: Multiplying Signed Fractions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.

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