Question

In 2 years a boy will be 3/4 as old as his sister. 2 years ago he was 2/3 as old as she. How old are they?

Answer 1

The boy is `10`
and his sister is `14`

Explanation:

Define
`color(white)("XXX")b`: boy's current age
`color(white)("XXX")g`: his sister's current age

We are told
[1]`color(white)("XXX")b+2 = 3/4(g+2)`
and
[2]`color(white)("XXX")b-2= 2/3(g-2)`

[1] can be simplified into standard form as
[3]`color(white)("XXX")4b-3g=-2`

[2] can be simplified into standard form as
[4]`color(white)("XXX")3b-2g=2`

Multiply [3] by 2 and [4] by 3
[5]`color(white)("XXX")8b-6g=-4`
[6]`color(white)("XXX")9b-6g=6`

Subtract [5] from [6]
[7]`color(white)("XXX")b=10`

Substitute `10` for `b` in [3]
[8]`color(white)("XXX")4(10)-3g=-2`

[9]`color(white)("XXX")3g = 42`

[10]`color(white)("XXX")g=14`