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?

i got 24.

Answer 1

let x = younger brother's age
y = older brother's age

their ratio is 2/3

\[\frac xy= \frac 23\]
so \[3x = 2y\]
\[\implies y = \frac{3x}{2}\]
in 8 years the ratio would be 3/4
\[\frac{x+ 8}{y + 8} = \frac 34\]
so
\[4(x+ 8) = 3(y+8)\]
\[\implies 4x + 32 = 3y + 24\]
\[\implies 4x - 3y = 24 - 32\]
\[\implies 4x - 3y = -8\]
if i substitute y = 3x/2
\[\implies 4x - 3(\frac{3x}{2}) = -8\]
multiply everything by 2
\[\implies 8x - 3(3x) = -16\]
\[8x - 9x = -16\]
\[-x = -16\]
\[x = 16\]
that is the age of the younger bro

Answer 2

i hope that made sense for you

Answer 3

woah thank you. i think i get it.