Question

Two numbers are in the ratio 7: 9. If each number is increased by 18, the ratio becomes 9 : 11 . Find the numbers

Answer 1

Solution :-

Let :-

The two numbers be x and y .

According to the first condition :-

$\longrightarrow \sf \dfrac{x}{y} = \dfrac{7}{9} \\\\\longrightarrow x = \dfrac{7}{9} y \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(1)$

According to the second condition :-

$\longrightarrow \sf \dfrac{x+18}{y+18} = \dfrac{9}{11} \\\\\longrightarrow 11(x+18)= 9 (y+18) \\\\\longrightarrow 11x+198 = 9y + 162 \\\\\longrightarrow 11x-9y = -36 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .................(2)$

Substitute the value of x in eq (2) :-

$\longrightarrow \sf 11 \times \dfrac{7}{9}y - 9y = - 36 \\\\\longrightarrow \dfrac{77y}{9} - 9y = - 36 \\\\\longrightarrow \dfrac{77y-81y}{9}= - 36 \\\\\longrightarrow -4y = -324 \\\\\longrightarrow \bf y = 81$

Substitute y = 81 in eq (1) :-

$\longrightarrow \sf x = \dfrac{7}{9} \times 81 \\\\\longrightarrow x = 7 \times 9 \\\\\longrightarrow \bf x = 63$

The two numbers are 63 and 81.