### Theory:

Situation $$1$$:

Consider the following situation.

Reena mother said, Reena completed $\frac{1}{3}$ of the homework exercises.

Ravi mother said, Ravi completed $\frac{1}{4}$ of the homework exercises.

Can you guess who completed more?

Let us express the ratios as a fraction and then find the equivalent fractions, until the denominators are the same, and compare the fractions with common denominators.

This situation can be solved by making the denominator the same.

 Reena's completed work Ravi's completed work $\frac{1}{3}×\frac{2}{2}=\frac{2}{6}$ $\frac{1}{4}×\frac{2}{2}=\frac{2}{8}$ $\frac{1}{3}×\frac{3}{3}=\frac{3}{9}$ $\underset{¯}{\frac{1}{4}×\frac{3}{3}=\frac{3}{12}}$ $\underset{¯}{\frac{1}{3}×\frac{4}{4}=\frac{4}{12}}$ $\frac{1}{4}×\frac{4}{4}=\frac{4}{16}$

Note that the fractions $\frac{1}{3}$ and $\frac{1}{4}$ have the same denominator $$12$$ with the values $\frac{4}{12}$ and $\frac{3}{12}$.

The numerator value of the fraction $\frac{1}{3}$ is $$4$$, and the numerator value of the fraction $\frac{1}{4}$ is $$3$$.

Thus, the numerator value is higher for $\frac{1}{3}$.

This implies Reena completed more work than Ravi.

Situation $$2$$:

A rope of $$8 m$$ is cut into two parts $$3 m$$ and $$5 m$$. This gives the ratio of the two pieces becomes $$3:5$$.

Important!
In general, a ratio $$a : b$$ is said to have a total of '$$a+b$$' parts in it.