Question

How do you simplify the fraction $\dfrac{{27}}{{30}}$?

Answer 1

Hint: In this question, we need to simplify the given fraction. i.e. we need to convert it into its smallest form. Simplifying a fraction requires dividing the numerator and denominator by the same number. This same number is nothing but the highest common factor of both the numbers. And then we simplify it in such a way that numerator and denominator should have 1 as only the common factor. This is the simplified form such that the numbers further cannot be divided by any other number.

Complete step by step solution:
Given the fraction of the form $\dfrac{{27}}{{30}}$
We are asked to find out the smallest form of this fraction.
First let us understand what actually the fractions means.
If a whole thing is divided into equal parts, then each part is said to be a fraction. In other words, a fraction is a part of the whole.
A fraction is formed up of two parts. One on top as numerator and one at the bottom as denominator which are separated by a horizontal line.
i.e. we write them as, $\dfrac{x}{y}$.
If we simplify fractions it becomes easier to deal with.
Now consider the given fraction $\dfrac{{27}}{{30}}$ …… (1)
Now let us find out the factors of numerator and denominator.
Factors of the numerator i.e. 27 are $1,3,9,27$
Factors of the denominator i.e. 30 are $1,2,3,5,6,10,15,30$
Note that the highest common factor of both numerator and denominator is 3.
So let us divide the fraction given in the equation (1) by 3, we get,
$ \Rightarrow \dfrac{{27 \div 3}}{{30 \div 3}} = \dfrac{9}{{10}}$
Now we obtained the new fraction $\dfrac{9}{{10}}$.
Note that there is no common factor in them as $\gcd (9,10) = 1$.
So this is a simplified answer.

Thus, the simplified form of the fraction $\dfrac{{27}}{{30}}$ is given by $\dfrac{9}{{10}}$.

Note :
Some facts in simplifying the fractions given below.
(1) We can simplify fractions if the numerator and denominator can both be divided by the same number.
(2) A common factor is a number that divides both the numerator and denominator without leaving any remainders.
(3) Most of the time we divide the fractions till we get 1 as the common factor for both the numerator and denominator.
(4) As we know that prime numbers are the special numbers which cannot be divided further. If a fraction contains prime numbers in both numerator and denominator, then it is already in its simplest form.
(5) Decimals are not used to simplify the fractions as it will become difficult to see the proportions and it will no longer stay in simpler form.