Question

Two numbers are in the ratio $5:6$. If $8$ is subtracted from each of the numbers, the ratio becomes $4:5$ find the numbers.

Answer 1

Ratio:

A ratio is a comparison of two or more quantities of the same units. If $\mathrm{p}$ and $\mathrm{q}$ are numbers, such that $\mathrm{q}\ne 0$, $\frac{\mathrm{p}}{\mathrm{q}}$ is the ratio of

$\mathrm{p}$ to $\mathrm{q}$. It can be written as $\mathrm{p}:\mathrm{q}$.

Calculating two unknown numbers:

The ratio of two numbers is given as $5:6$.

Consider two numbers $5\mathrm{x}$ and $6\mathrm{x}$, where $\mathrm{x}$ is a common multiple of two numbers.

Also, the ratio of two numbers is given as $4:5$ when $8$ is subtracted from each of the numbers $5\mathrm{x}$ and $6\mathrm{x}$.

$$\begin{gathered} \Rightarrow \frac{5\mathrm{x}-8}{6\mathrm{x}-8}=\frac{4}{5} \\ \Rightarrow 5\left(5\mathrm{x}-8\right)=4\left(6\mathrm{x}-8\right) \\ \Rightarrow 25\mathrm{x}-40=24\mathrm{x}-32 \\ \Rightarrow 25\mathrm{x}-24\mathrm{x}=40-32 \\ \Rightarrow \mathrm{x}=8 \end{gathered}$$

Substituting the value of $\mathrm{x}$ in two numbers :

$$\begin{gathered} 5\mathrm{x}=5\times 8 \\ \Rightarrow 5\mathrm{x}=40 \end{gathered}$$

$$\begin{gathered} 6\mathrm{x}=6\times 8 \\ \Rightarrow 6\mathrm{x}=48 \end{gathered}$$

Hence the two numbers are $40$ and $48$.