Question

Petra is 4 times as old as Philippa. In 8 years time she will be 5 times as old as Philippa. How old are Philippa and Petra?

Answer 1

The question as given has no positive solutions.

If the `4` and `5` were the other way round then Petra's current age is `120` and Philippa's `24`.

Explanation:

The question as given has no positive solutions.

Let Petra's current age be `x` and Philippa's `y`.

We are given:

`x = 4y`

`x+8 = 5(y + 8) = 5y + 40`

Subtract the first of these equations from the second to get:

`8 = x+8 - x = 5y+40 - 4y = y + 40`

Subtract `40` from both sides to get:

`y = -32`

Then

`x = 4y = 4(-32) = -128`

So Petra is `-128` and Philippa is `-32`

Alternative problem

Suppose the `4` times and `5` times were supposed to be the other way round.

Then we are given:

`x = 5y`

`x + 8 = 4(y+8) = 4y + 32`

Subtract the second equation from the first to get:

`-8 = x - (x + 8) = 5y - (4y + 32) = y - 32`

Add `32` to both ends to get:

`y = -8 + 32 = 24`

Then

`x = 5y = 5*24 = 120`

So Petra is `120` and Philippa is `24`

Answer 2

Petra = -32, Philippa = 94
Given that initial condition year is `t_0`, Petra had not been born. She would be in 32 years that is, `t_32`. Consequently Petra's age is negative.

Explanation:

I gave the wrong answer originally. Submission deleted!
Hopefully I will not get interrupted this time!!! Wrong, 4th attempt!

Let current time be `t_0`
Let time in 8 years be `t_8`
Let Petra be `"Pe"`
Let Philippa be `"Ph"`

At `t_o -> "Pe" = 4"Ph"`
Then `"Pe - 4"Ph" = 0`.................. ( 1 )

At `t_8 -> "Pe" + 8 = 5("Ph" + 8)`
Then `"Pe" -5"Ph" = 40 - 8 = 32`....... ( 2 )

Subtract ( 1 ) from ( 2 ) giving:

`-"Ph" = 32`
So `"Ph" = -32` .................... ( 3 )

( at first this looks as though it is not possible. However if you view negative age as to be born in the future then it works. So -32 means that `"Ph is to born at " t_32`
If at `t_0 " Pe"` is four times older then, stepwise we have from `"Ph's"` age:

(-32), 32, 32, 32

So `"Pe's"` age at `t_0` is `32 + 32 + 32= 94` because `Ph's` age is 32 years the other side of 0 giving a total of `4 times 32 = 128` years difference. ( `94 - (-32)) = 128`