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Multiplying Fractions


A fraction is the number expressed in form of $\frac{p}{q}$, where ā€˜pā€™ and ā€˜qā€™ are whole numbers and q $\neq$ 0. Here we are going to learn about multiplying fractions, we mean to find the product of two or more given fractions.

When we multiply the two fractions, we need to multiply the numerator with the numerator and the denominator with the denominator. The resultant fraction we get is the product of the two given fractions. Now we will convert the resultant fraction into its lowest form.

Example: $\frac{4}{7}$ * $\frac{3}{8}$ = $\frac{4*3}{7*8}$ = $\frac{12}{56}$ = $\frac{6}{28}$.
Also we must remember that if we multiply fraction by its inverse, then the product of the given fraction is always 1.

Example: $\frac{3}{7}$ * $\frac{7}{3}$ = 1

Example: $\frac{2}{5}$ When any fraction is multiplied by 0, the result is always 0. * 0 = 0
If the fraction is multiplied by 1, the result is the given fraction itself. So we say that 1 is the multiplicative identity.

Example: $\frac{2}{5}$ * 1 = $\frac{2}{5}$

Rules for Multiplying Fractions

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To multiply fractions follow the rules given below.
  1. If possible simplify the fractions.
  2. Multiply the numerators of the fractions to get the new numerators.
  3. Multiply the denominators of the fractions to get the denominators.
  4. Place the product of the numerators over the products of the denominators.
  5. The resulting fractions needs to be simplified if possible.

Multiplying Negative Fractions

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Fractions can be positive or negative. Negative fractions are dealt in the same way as whole negative numbers, and can also be calculated on a number line. If the two fractions are of the same sign the answer is always positive. If the signs are different, the result is negative.
Example 1: Multiply $\frac{2}{3}$ and $\frac{-1}{2}$

Given: $\frac{2}{3}$ and $\frac{-1}{2}$

$\times$ $\frac{-1}{2}$

= $\frac{-2}{6}$

or $\frac{-1}{3}$ (By simplifying)

Therefore, the answer is $\frac{-1}{3}$

Example 2: Multiply $\frac{-7}{5}$ and $\frac{-5}{2}$

Given: $\frac{- 7}{5}$ and $\frac{-5}{2}$

$\frac{-7}{5}$ $\times$ $\frac{-5}{2}$

$\frac{(-7 * -5)}{5*2}$

= $\frac{+35}{10}$

The answer can be simplified further.
Therefore, the answer is $\frac{7}{2}$

Multiplying Fractions with Whole Numbers

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To multiply the whole number with fraction, multiply the numerator of the fraction with whole number and place the result over the denominator.

Example: Multiply $\frac{2}{5}$ with whole number 6.
Multiply the numerators of the fractions with whole number.
6 x $\frac{2}{5}$ = $\frac{6*2}{5}$

Place the result over the denominator.
We get, $\frac{12}{5}$

Therefore, the solution is $\frac{12}{5}$

Multiplying Fractions with Variables

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Multiplying fractions with variables is explained below with the help of few examples.

Example 1: Multiply $\frac{5x^{4}}{3y^{2}}$ * $\frac{7y}{2x^{2}}$

Given: $\frac{5x^{4}}{3y^{2}}$*$\frac{7y}{2x^{2}}$


= $\frac{35x^{2}}{6y}$

Example 2: $\frac{x^{2}-y^{2}}{6x - 6y}$ * $\frac{3x}{(5x+5x)}$
Given: $\frac{x^{2}-y^{2}}{6x - 6y}$ * $\frac{3x}{(5x+5y)}$

= $\frac{(x+y)(x-y)}{6(x-y)}$ * $\frac{3x}{5(x+y)}$

= $\frac{x}{10}$

Multiplying Mixed Fractions

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For multiplying fraction we need to follow the simple steps given below.
  1. First change the mixed fractions into improper fractions.
  2. Multiply the numerator with numerator and denominator with denominator of first fraction to second one.
  3. Convert the result to a mixed fraction. Simplify the answer if needed.

Example 1: Multiplying the mixed numbers 2$\frac{1}{3}$ and 6$\frac{1}{7}$.


Step 1: Convert the mixed numbers 2$\frac{1}{3}$ to improper fraction, we get

2$\frac{1}{3}$ = $\frac{7}{3}$

Step 2: Convert the mixed numbers 6 $\frac{1}{7}$ to improper fraction, we get

6 $\frac{1}{7}$ = $\frac{43}{7}$

Step 3: Multiplying the improper fractions $\frac{7}{3}$ and $\frac{43}{7}$ , we get

$\frac{7}{3}$ x $\frac{43}{7}$ = $\frac{301}{21}$

Step 4: Convert the improper fraction $\frac{301}{21}$ to mixed number.

Mixed number $\frac{301}{21}$ = 14 $\frac{7}{21}$

Answer: 14$\frac{7}{21}$