How do you simplify $\dfrac{{\dfrac{3}{4}}}{3}$?

Hint: In this question, we need to simplify the given fraction. i.e. to convert the given fraction into its smallest form. Now to simplify the given fraction, we try to convert some of the terms, so that it becomes easier to solve. We use the fact that dividing by 3 is the same as multiplying by $\dfrac{1}{3}$. So replace 3 by $\dfrac{1}{3}$ in the denominator and simplify. Try to cancel out the terms and obtain a simplified version of the given fraction.

Complete step by step solution:
Given the fraction $\dfrac{{\dfrac{3}{4}}}{3}$ …… (1)
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 let us consider the given fraction mentioned in the equation (1).
Firstly, let us look at the number 3 in the denominator.
We know that dividing by a number is the same as multiplying by the reciprocal of the same number.
So dividing by 3 is the same as multiplying by $\dfrac{1}{3}$.
So replace 3 in denominator, we get,
$ \Rightarrow \dfrac{{\dfrac{3}{4}}}{3} = \dfrac{3}{4} \times \dfrac{1}{3}$
Look at the R.H.S. of the equation. If we carefully observe, we can cancel out some terms.
Since 3 is common in both numerator and denominator, cancelling it we get,
$ \Rightarrow \dfrac{{\dfrac{3}{4}}}{3} = \dfrac{1}{4} \times 1$
This can be written as,
$ \Rightarrow \dfrac{{\dfrac{3}{4}}}{3} = \dfrac{1}{4}$

Hence the simplified form of the fraction $\dfrac{{\dfrac{3}{4}}}{3}$ is given by $\dfrac{1}{4}$.

Note: Remember that dividing by a number is the same as multiplying by the reciprocal of the same number.
 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.