Two numbers are in the ratio 4 : 7. If thrice the larger be added to twice the smaller, the sum is 59. Find the numbers.

Let two numbers be x and y, where x is the larger and y is the smaller number.

Given,

Two numbers are in the ratio 4 : 7.

$\therefore \dfrac{y}{x} = \dfrac{4}{7} \\[1em] \Rightarrow y = \dfrac{4x}{7} …….(1)$

Given,

If thrice the larger be added to twice the smaller, the sum is 59.

∴ 3x + 2y = 59

$\Rightarrow 3x + 2 \times \dfrac{4x}{7} = 59 \\[1em] \Rightarrow 3x + \dfrac{8x}{7} = 59 \\[1em] \Rightarrow \dfrac{21x + 8x}{7} = 59 \\[1em] \Rightarrow \dfrac{29x}{7} = 59 \\[1em] \Rightarrow x = \dfrac{59 \times 7}{29} \\[1em] \Rightarrow x = 14\dfrac{7}{29}.$

Substituting value of x in equation (1), we get :

$\Rightarrow y = \dfrac{4}{7} \times 14\dfrac{7}{29} \\[1em] = \dfrac{4}{7} \times \dfrac{413}{29} \\[1em] = \dfrac{4 \times 59}{29} \\[1em] = 8\dfrac{4}{29}.$

Hence, numbers are $8\dfrac{4}{29} \text{ and } 14\dfrac{7}{29}$.