Question

The ages of two brothers are in the ratio two to three, but in eight years, the ratio of their ages will be three to four. What is the age of the younger brother?

Answer 1

x = older brother

y = younger brother

Equation 1: x/y = 2/3 ⇒ x = 2y/3

Equation 2: x + 8/y + 8 = 3/4

( \frac{2y}{3} + 8) / y +8 =  \frac{3}{4}

(2y + 24)/3 = y + 8 =  \frac{3}{4}  

2y + 24 = 3 (y + 8) =  \frac{3}{4}

2y + 24 = 3y + 24 =  \frac{3}{4}

4 (2y + 24) = 3 (3y + 24)

8y + 96 = 9y + 72

y = 24

The age of the older brother is 24.

x =  \frac{2y}{3} = \frac{2*24}{3}

x =  \frac{48}{3}

x = 16

The age of the youngest brother is 16.

Answer 2

X = older brother
y = younger brother 

Equation 1: x/y = 2/3 ⇒ x = 2y/3
Equation 2: x + 8/y + 8 = 3/4

$( \frac{2y}{3} + 8) / y +8 = \frac{3}{4}$
$(2y + 24)/3 = y + 8 = \frac{3}{4}$
$2y + 24 = 3 (y + 8) = \frac{3}{4}$
$2y + 24 = 3y + 24 = \frac{3}{4}$
$4 (2y + 24) = 3 (3y + 24)$
$8y + 96 = 9y + 72$
$y = 24$

The age of the older brother is 24.

$x = \frac{2y}{3} = \frac{2*24}{3}$
$x = \frac{48}{3}$
$x = 16$

The age of the youngest brother is 16.