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Equivalent Fractions Simplest Form

Equivalent Fractions can be cut down when the upper and lower value have a same factor in them. If the upper and lower value has a same factor, then we can withdraw the factors out.

For example, in the fraction $\frac{6}{9}$, 3 is the common factor of both 6 and 9.

`(6)/(9)` = `(2xx 3)/(3 xx3)`

In this, we can simplify the fractions by canceling the 3 from both the upper and lower of the fraction. Canceling means dividing both the upper and lower by the same number.

`(6)/(9)` = `(2xx 3)/(3xx 3)` = `(2)/(3)` is equivalent to `(6)/(9)` = `(6 -:3)/(9 -: 3)`

To simplify a fraction we want to find all the common factors between upper and lower and to eliminate them. The easiest way to eliminate the common factors for upper and lower factors is to find the prime factors of each of them and then cancel them out.Equivalent Fractions Simplest Form

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Steps to Convert Equivalent Fractions to Simplest Form

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  • Write the prime factorization of both the upper and lower.

  • Rewrite the fractions so that the upper and lower are written as the product of their prime factors.

  • Withdraw out any common prime factors.

  • Multiply together any remaining factors in the upper and lower.

Equivalent Fractions and Simplest Form

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Equivalent fractions : If the numerator and denominators of two different fractions are the multiples of a number, then those fractions are called as equivalent fractions.
For example: $\frac{2}{3}$ and $\frac{6}{9}$ are equivalent fractions.
Because, the numerators 2 and 6 are multiples of 2 and the denominators 3 and 9 are multiples of 3.

Simplest form of fractions : A fraction is said to be in simplest form, only when there is no common factors between the numerator and denominator.

For example:

$\frac{2}{3}$ is in simplest form.
$\frac{2}{6}$ is not the simplest form because the numerator 2 and the denominator 6 have 2 as a common factor.

Solved Examples

Question 1: Check whether the following fractions are equivalent fractions.
$\frac{1}{2}$ , $\frac{2}{6}$ , and $\frac{2}{9}$
Solution:
To find whether the given fractions are equivalent, we need to find the factors of the numerators and denominators of all fractions.

$\frac{1}{2}$ can be written as $\frac{1}{2}$

$\frac{2}{6}$ can be written as $\frac{1}{2}$ $\frac{1}{3}$

$\frac{2}{9}$ can be written as ($\frac{1}{3}$)($\frac{2}{3}$)

There are no common terms in the fractions, so the given fractions are not equivalent.


Question 2: Find the simplest form of the fraction $\frac{25}{265}$
Solution:
We know that, the fraction $\frac{25}{265}$ is not in the simplest form because the numerator and denominator has 5 as common factor.

To find the simplest form of $\frac{25}{265}$ , we need to find the prime factorization of both numerator and denominator.

So, 25 = 5 x 5

265 = 5 x 53

So $\frac{25}{265}$ becomes, $\frac{(5)(5)}{(5)(53)}$ = $\frac{5}{53}$

So, $\frac{5}{53}$ is the simplest form of $\frac{25}{265}$ .


Examples of Solving Equivalent Fractions to Simplest Form

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Below you could see the example of solving equivalent fractions to simplest form

Solved Example

Question: Find the simplest form of the equivalent fraction `20/32`.
Solution:
 
The simplest form of `20/32` is `5/8`. Steps to solve this equivalent fraction:

1.  Write the prime faction of both the upper and lower
  
    The prime factors of 20 are 2,2 and 5.
    The prime factors of 32 are 2, 2, 2,2 and 2.

2. Rewrite the fraction so that the upper and lower are written as the product of their prime factors.

`(20)/(32)` = `(2 xx2xx 5)/(2 xx 2 xx 2 xx 2xx2)`

3. Withdraw out any common prime factors.

Cancel out a 2 from both the numerator and the denominator.

`(20)/(32)` = `( 5)/(2 xx 2 xx 2 )`

4. Multiply together any remaining factors in the upper and lower.

`(20)/(32)` = `( 5)/(2 xx 2 xx 2)` = `(5)/(8)`

The fraction `(20)/(32)` can be simplified to `(5)/(8)` .
 

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