Question

Two numbers are in the ratio of $3:4$. If 4 be added to each, the ratio becomes $4:5$. Find the numbers.

Answer 1

Step1: Find a relation between given two numbers:

Let the given numbers are $x$ and $y$.

$$\begin{gathered} \Rightarrow \frac{x}{y}=\frac{3}{4} \\ \Rightarrow x=\frac{3}{4}y\dots \dots Equation\left(1\right) \end{gathered}$$

Step2: Find the value of $y$:

Add 4 to both $x$ and $y$ to obtain the ratio equivalent to $4:5$ as $x+4:y+4$.

$$\begin{gathered} \Rightarrow \frac{x+4}{y+4}=\frac{4}{5} \\ \Rightarrow \frac{{\displaystyle \frac{3}{4}}y+4}{y+4}=\frac{4}{5}(Fromequation(1\left)\right) \\ \Rightarrow \frac{{\displaystyle \frac{3y+16}{4}}}{y+4}=\frac{4}{5} \\ \Rightarrow \frac{3y+16}{4(y+4)}=\frac{4}{5} \\ \Rightarrow \frac{3y+16}{4y+16}=\frac{4}{5} \\ \Rightarrow 5(3y+16)=4(4y+16)(Cossmultiplyingbothsides) \\ \Rightarrow 15y+80=16y+64 \\ \Rightarrow 16y-15y=80-64(Separatingthevariableandcons\tan tterms) \\ \therefore y=16 \end{gathered}$$

Step3: Find the value of $x$:

Substitute $y=16$ in the equation $\left(1\right)$ and simplify to obtain the value of $x$.

$$\begin{gathered} x=\frac{3}{4}\times 16 \\ \Rightarrow x=3\times 4 \\ \therefore x=12 \end{gathered}$$

Final Answer: The numbers are 12 and 16.