Question

Two numbers are in the ratio $$8 : 13$$. If 14 is added to each of the numbers, the ratio becomes $$2 : 3$$. Find the numbers.

Answer 1

Let the two numbers be $$8x$$ and $$13x$$.

According to the problem: $$\frac{8x + 14}{13x + 14} = \frac{2}{3}$$ 

Cross-multiply: $$3(8x + 14) = 2(13x + 14)$$

$$24x + 42 = 26x + 28$$ 

Solve for $$x$$: $$26x - 24x = 42 - 28$$

$$2x = 14 \implies x = 7$$ 

Calculate the numbers:

First number = $$8 \times 7 = 56$$

Second number = $$13 \times 7 = 91$$

Therefore, the numbers are 56 and 91.