Question

Having some trouble with these questions, how do I work them out?

Answer 1

5.

Let x be the age of the father and y be the age of the son. As of today, he's 3 times older, so we have $x=3y$

10 years ago their ages were, respectively, x-10 and y-10, and the father was 5 times older: $x-10 = 5(y-10)$

So, we have the system

$\begin{cases} x=3y\\x-10=5(y-10)\end{cases}$

Using the first equation, we can substitute every occurrence of "x" with "3y" in the second equation:

$3y-10=5(y-10) \iff 3y-10=5y-50 \iff 2y = 40 \iff y=20$

So, the son is 20 years old, which means that the father is 60 years old.

Indeed, 10 years ago they were 10 and 50 years old, so the father was 5 times older.

6.

Let x be the age of the grandfather and y the age of the granddaughter. We know that the grandfather is 10 times older: $x=10y$

He also is 54 years older: $x=y+54$

Again, if we substitute x=10y in the second equation we have

$10y=y+54 \iff 9y=54 \iff y=6$