Daniel, my son, is exactly one fifth of my age. in 21 years time, i will be exactly twice his age. my wife is exactly seven times older than my daughter, jessica. in 8 years time, my wife will be three times older than jessica. how old are daniel and jessica now?

Let daniel's age = d and let jessica's age = j
let your current age = x and your wife's current age = y

x = 5d (given in question)
∴ x - 5d = 0

x + 21 = 2(d+21) (given in question)
∴ x + 21 = 2d + 42
∴ x - 2d = 21

We can solve these two rearranged equations simultaneously by multiplying the first equation by -1 and adding them. This gives us the following:

3d = 21
∴ d = 7

This means that daniel is currently 7 and (if we substitute d = 7 into one of the equations) you are 35. We use a similar method for your wife's and jessica's current ages.

y = 7j (given in question)
∴ y - 7j = 0

y + 8 = 3(j + 8)
∴y + 8 = 3j + 24
∴ y - 3j = 16

If we use a similar method of elimination we get this:

4j = 16
∴ j = 4

Hence, from this we can concur that daniel is 7 and jessica is 4.