Multiplication of Fractions

When multiplying fractions, we multiply the numerators together to obtain the numerator of the result, and multiply the denominators together to obtain the denominator of the result. After this, we simplify the fraction by eliminating any common factors from the numerator and the denominator.

Alternatively, we can first cancel out any common factors between the numerators and the denominators of the fractions before performing the multiplication. This method can save time and effort.

For example:

\frac { 2 }{ 5 } \times \frac { 15 }{ 17 } =\frac { 2\times 15 }{ 5\times 17 } =\frac { 30 }{ 85 }

Both 30 and 85 have a common factor of 5. We can simplify the fraction by dividing both the numerator and the denominator by 5, resulting in \frac { 6 }{ 17 }.

Examples

Here are some more examples:

1. \frac { 5 }{ 8 } \times \frac { 2 }{ 15 }

First, cancel out common factors:

\frac { 5 }{ 8 } \times \frac { 2 }{ 15 } =\frac { 5 }{ 2\times 4 } \times \frac { 2 }{ 5\times 3 } =\frac { 1 }{ 4 } \times \frac { 1 }{ 3 }\frac { 5 }{ 8 } \times \frac { 2 }{ 15 } =\frac { 5 }{ 2\times 4 } \times \frac { 2 }{ 5\times 3 } =\frac { 1 }{ 4 } \times \frac { 1 }{ 3 }

Now, multiply the simplified fractions:

=\frac { 1 }{ 12 }=\frac { 1 }{ 12 }

2. \frac { 6 }{ 13 } \times \frac { 5 }{ 18 }

Cancel out common factors:

\frac { 6 }{ 13 } \times \frac { 5 }{ 6\times 3 } =\frac { 5 }{ 13\times 3 }\frac { 6 }{ 13 } \times \frac { 5 }{ 6\times 3 } =\frac { 5 }{ 13\times 3 }

Multiply the simplified fractions:

=\frac { 5 }{ 39 }=\frac { 5 }{ 39 }

3. \frac { 3 }{ 10 } \times \frac { 5 }{ 9 } \times \frac { 12 }{ 13 }

Write numbers as common factors:

\frac { 3 }{ 10 } \times \frac { 5 }{ 9 } \times \frac { 12 }{ 13 } = \frac { 3 }{ 5 \times 2 } \times \frac { 5 }{ 3 \times 3} \times \frac { 6 \times 2 }{ 13 }\frac { 3 }{ 10 } \times \frac { 5 }{ 9 } \times \frac { 12 }{ 13 } = \frac { 3 }{ 5 \times 2 } \times \frac { 5 }{ 3 \times 3} \times \frac { 6 \times 2 }{ 13 }

Cancel common factors in the numerators and denominators:

\frac { 3 }{ 5 \times 2 } \times \frac { 5 }{ 3 \times 3} \times \frac { 6 \times 2 }{ 13 } = \frac { 6 }{3 \times 13}\frac { 3 }{ 5 \times 2 } \times \frac { 5 }{ 3 \times 3} \times \frac { 6 \times 2 }{ 13 } = \frac { 6 }{3 \times 13}

We cancelled 3, 5 and 2, as they are present in both the numerator and denominator of the fractions we are multiplying. This leaves us with a simplified product of = \frac { 6 }{ 39 }= \frac { 6 }{ 39 }

Simplify by dividing the numerator and denominator by 3.

= \frac { 6 \div 3 }{ 39 \div 3 } = \frac { 2 }{ 13}= \frac { 6 \div 3 }{ 39 \div 3 } = \frac { 2 }{ 13}

4. 1\frac { 1 }{ 2 } \times 2\frac { 2 }{ 5 } \times 3\frac { 2 }{ 3 }

1. Convert the mixed numbers to improper fractions:

1\frac { 1 }{ 2 } =\frac { 3 }{ 2 }1\frac { 1 }{ 2 } =\frac { 3 }{ 2 }, 2\frac { 2 }{ 5 } =\frac { 12 }{ 5 }2\frac { 2 }{ 5 } =\frac { 12 }{ 5 }, and 3\frac { 2 }{ 3 } =\frac { 11 }{ 3 }3\frac { 2 }{ 3 } =\frac { 11 }{ 3 }

2. Rewrite the product using the improper fractions:

\frac { 3 }{ 2 } \times \frac { 12 }{ 5 } \times \frac { 11 }{ 3 }\frac { 3 }{ 2 } \times \frac { 12 }{ 5 } \times \frac { 11 }{ 3 }

3. Factor the numerators and denominators to identify common factors:

\frac { 3 }{ 2 } \times \frac { 6 \times 2 }{ 5 } \times \frac { 11 }{ 3 }\frac { 3 }{ 2 } \times \frac { 6 \times 2 }{ 5 } \times \frac { 11 }{ 3 }

4. Cancel common factors that appear in both the numerators and denominators. In this case, cancel out the 3 and 2:

*** QuickLaTeX cannot compile formula:
\frac {\cancel{3}}{ \cancel{2} } \times \frac { 6 \times \cancel{2} }{ 5 } \times \frac { 11 }{ \cancel{3} }
*** Error message:
Undefined control sequence \cancel.
leading text: $\frac {\cancel{3}}{ \cancel{2} }

5. Multiply the remaining numerators and denominators:

\frac { 1 }{ 1 } \times \frac { 6 }{ 5 } \times \frac { 11 }{ 1 } = \frac { 6 \times 11 }{ 5 }\frac { 1 }{ 1 } \times \frac { 6 }{ 5 } \times \frac { 11 }{ 1 } = \frac { 6 \times 11 }{ 5 }

6. Calculate the resulting fraction:

\frac { 6 \times 11 }{ 5 } = \frac { 66 }{ 5 }\frac { 6 \times 11 }{ 5 } = \frac { 66 }{ 5 }

7. Convert the improper fraction back to a mixed number:

\frac { 66 }{ 5 } = 13\frac { 1 }{ 5 }\frac { 66 }{ 5 } = 13\frac { 1 }{ 5 }

So, the simplified product of the given mixed numbers is 13\frac{1}{5}13\frac{1}{5}.

5. 1\frac { 1 }{ 2 } \times 2\frac { 1 }{ 3 } \times 3\frac { 1 }{ 4 } \times 4\frac { 1 }{ 5 }

1. Convert the mixed numbers to improper fractions:

1\frac { 1 }{ 2 } =\frac { 3 }{ 2 }1\frac { 1 }{ 2 } =\frac { 3 }{ 2 }, 2\frac { 1 }{ 3 } =\frac { 7 }{ 3 }2\frac { 1 }{ 3 } =\frac { 7 }{ 3 }, 3\frac { 1 }{ 4 } =\frac { 13 }{ 4 }3\frac { 1 }{ 4 } =\frac { 13 }{ 4 }, and 4\frac { 1 }{ 5 } =\frac { 21 }{ 5 }4\frac { 1 }{ 5 } =\frac { 21 }{ 5 }

2. Write the product using the improper fractions:

\frac { 3 }{ 2 } \times \frac { 7 }{ 3 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 }\frac { 3 }{ 2 } \times \frac { 7 }{ 3 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 }

3. Notice that there are common factors between the numerators and denominators that can be canceled out:

\frac { 1 }{ 2 } \times \frac { 7 }{ 1 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 }\frac { 1 }{ 2 } \times \frac { 7 }{ 1 } \times \frac { 13 }{ 4 } \times \frac { 21 }{ 5 }

4. Multiply the remaining numerators and denominators:

\frac { 1\times 7\times 13\times 21 }{ 2\times 1\times 4\times 5 }\frac { 1\times 7\times 13\times 21 }{ 2\times 1\times 4\times 5 }

5. Calculate the resulting fraction:

\frac { 3\times 7\times 13\times 21 }{ 2\times 1\times 4\times 5 } = \frac { 1911 }{ 40 }\frac { 3\times 7\times 13\times 21 }{ 2\times 1\times 4\times 5 } = \frac { 1911 }{ 40 }

6. Convert the improper fraction back to a mixed number:

\frac { 1911 }{ 40 } = 47\frac { 31 }{ 40 }\frac { 1911 }{ 40 } = 47\frac { 31 }{ 40 }

So, the simplified product of the given mixed numbers is 47\frac{31}{40}47\frac{31}{40}.

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