Question

if 2 is added to each of the two numbers the ratio becomes 1 : 2 and when 4 is subtracted from each of these the ratio becomes 5:11 find the numbers

Answer 1

Answer:

Let the two numbers be x and y respectively.

First Condition ;

When 2 is added to both, the ratio becomes 1 : 2.

$\frac{x + 2}{y + 2} = \frac{1}{2} \\ \\ 2(x + 2) = 1(y + 2) \\ \\ 2x + 4 = y + 2 \\ \\ 2x - y = 2 - 4 \\ \\ 2x - y = - 2 \\ \\ y = 2x + 2.........(i)$

Second condition;

When 4 is subtracted from both numbers, the ratio becomes 5:11

$\frac{x - 4}{y - 4} = \frac{5}{11} \\ \\ 5(y + 4) = 11(x - 4) \\ \\ 5y + 20 = 11x - 44 \\ \\ 5y - 11x = - 44 - 20 \\ \\ 5y - 11x = - 64 \\ \\ 5y = - 64 + 11x.......(ii)$

Putting the value of (i), in (ii),

$5y = - 64 + 11x \\ \\ 5(2x + 2) = - 64 + 11x \\ \\ 10x + 10 = - 64 + 11x \\ \\ 10x - 11x = - 64 - 10 \\ \\ - x = - 74 \\ \\ \bf \therefore x = 74$

Substituting the value of x in (i)

$y = 2x + 2 \\ \\ y = 2 \times 74 + 2 \\ \\ y = 148 + 2 \\ \\ \bf \therefore y = 150$

Hence, the numbers are 74 and 150.