Question

Solve each of the following problems by using two variables.
(i) Two numbers are in the ratio 3 : 4. If 4 is added to each, the ratio becomes 4 : 5, Find the numbers.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{first}\mathrm{number}\mathrm{be}x\mathrm{and}\mathrm{the}\mathrm{second}\mathrm{number}\mathrm{be}y. \\ \mathrm{Their}\mathrm{ratio}\mathrm{is}3:4.\left(\mathrm{Given}\right) \\ \therefore \frac{x}{y}=\frac{3}{4} \\ \Rightarrow 4x=3y...\left(\mathrm{i}\right) \\ \mathrm{If}4\mathrm{is}\mathrm{added}\mathrm{to}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{number},\mathrm{then}\mathrm{the}\mathrm{ratio}\mathrm{becomes}4:5. \\ \therefore \frac{x+4}{y+4}=\frac{4}{5} \\ \Rightarrow 5x+20=4y+16 \\ \Rightarrow 5x-4y=-4 \\ \Rightarrow x=\frac{4y-4}{5}....\left(\mathrm{ii}\right) \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\mathrm{equation}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ 4\times \left(\frac{4y-4}{5}\right)=3y \\ \Rightarrow 16y-16=15y \\ \Rightarrow y=16 \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\mathrm{equation}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get}: \\ x=\frac{4\times 16-4}{5} \\ \Rightarrow x=12 \\ \text{The f}\mathrm{irst}\mathrm{number}\mathrm{is}12\mathrm{and}\mathrm{second}\mathrm{number}\mathrm{is}16. \end{gathered}$$