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# Reducing Fractions

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 Sub Topics Fractions are of the form $\frac{a}{b}$ where a and b are whole numbers and b$\neq$ 0. If the fractions in their lowest form are equal then they are said to be equivalent fractions. To reduce fractions first find highest common factor among the given fractions.Divide numerator and denominator by the highest common multiple. All the equivalent fractions, when converted into their lowest form, results in the same fraction.

## Reducing Fractions to the Lowest Terms

To reduce a fraction to its simplest form, numerator and denominator should have no common factors. When a fraction is simplified it is easier to add, subtract, multiply and divide.
Given below are the steps to reduce a fraction to its lowest terms.
1. List the prime factors of numerator and denominator.
2. Identify the common factors.
3. Cancel out the common factors of numerator and denominator.
Example: Simplify $\frac{4}{8}$ to its lowest terms.

Solution: Given fraction $\frac{4}{8}$
Step 1:
Prime factors of 4: 2 * 2
Prime factors of 8 : 2 * 2 * 2
We see the common factor is 4.

Step 2:
$\frac{4}{8}$ = $\frac{2*2}{2*2*2}$

Cancel out the common terms in both the numerator and the denominator.

= $\frac{\not{2}*\not{2}}{\not{2}*\not{2}*{2}}$

= $\frac{1}{2}$

Therefore the simplified solution for the given fraction is $\frac{1}{2}$.

## Reducing Fractions with Variables

Given below is an example explaining reducing fractions with variables.

Example: Simplify $\frac{49x^{2}y}{63xy^{2}z^{2}}$
Solution:
Step 1:
Prime factors of 49 : 7 * 7
Prime factors of 63 : 3 * 3 * 7
We see the common factor is 7.

Step 2:
$\frac{49x^{2}y}{63xy^{2}z^{2}}$ = $\frac{7*7*x*x*y}{3*3*7*x*y*y*z*z}$

Cancel out the common terms in both the numerator and the denominator.

= $\frac{\not{7}*7*\not{x}*x*\not{y}}{3*3*\not{7}*\not{x}*\not{y}*y*z*z}$

= $\frac{7x}{9yz^{2}}$

Therefore the simplified solution for the given fraction is $\frac{7x}{9yz^{2}}$.

## Reducing Improper Fractions

If the numerator value is greater than the denominator value, then the fraction is an improper fraction. The value of this fraction will always be greater than 1.

Example:
$\frac{7}{6}$, $\frac{9}{4}$, $\frac{8}{3}$

Given below is an example explaining how to reduce improper fractions.

Example 1: Simplify $\frac{24}{8}$
24 can be written as : 2 * 2 * 2 * 3
8 can be written as: 2 * 2 * 2
Step 2:
$\frac{24}{8}$ = $\frac{2*2*2*3}{2*2*2}$

Cancel out the common terms in both the numerator and the denominator.

= $\frac{\not{2}*3*\not{2}*\not{2}}{\not{2}*\not{2}*\not{2}}$
= 3
Therefore the simplified solution for the given fraction is 3.

## Reducing Fractions Examples

Example 1: Simplify the fraction $\frac{20}{30}$
Solution:
$\frac{20}{30}$ can be written as $\frac{2*2*5}{2*3*5}$

= $\frac{20}{30}$ = $\frac{2*2*5}{2*3*5}$

Cancelling the common terms we get
= $\frac{2}{3}$

Therefore for the given fraction $\frac{20}{30}$ the simplified fraction is $\frac{2}{3}$.

Example 2: Simplify the fraction $\frac{42}{56}$
Solution:
$\frac{42}{56}$ = $\frac{2*3*7}{2*2*2*7}$

By cancelling the common terms we get, $\frac{3}{4}$

Therefore for the given fraction $\frac{42}{56}$ the simplified fraction is $\frac{3}{4}$.