Question

Paul is one year older than his wife and they have two children whose ages are also one year apart.Paul noticed that on his birthday in 2011, the product of his age and his wife's age plus the sum of his children's ages is 2011. What would have been the result if he had done this calculation thirteen years before?

Answer 1

let's let P = Paul's age, W = wife's age, X and Y = the children's ages. To simplify the calculations, let's let X = the older child.

The children's ages are two years apart, so X-Y = 1
Paul is 1 year older than his wife, so P = W + 1
In 2001, the product of his age and his wife's age plus the sum of his children's ages is 2011 ---> PW + X + Y = 2011

Answer 2

That's all I have right now, I need to think more about this problem.

Answer 3

(P-13)(W-13) + (X-13) + (Y-13) = ?
(PW)^2 -13P - 13W + 13^2 + X + Y – 26 = ?

Yeah I've tried playing around w/ the equations for a bit, couldn't manage to get them down to a single unknown.

@surjithayer @mhanifa would you mind taking a look at this question when you get a chance

Answer 4

You can simplify it further.
Since P = W + 1 and X = Y + 1 we have it as:
W(W + 1) + X + X + 1= 2011
W^2 + W + 2X - 2010 = 0

Answer 5

Now try to find W.
The discriminant:
D = 1 - 8X + 8040 should be a square of an odd number if we are looking for integer solution.
We can consider the value of X between 1 and 100 and estimate the value of D.

Answer 6

There are no much choices: D = 87^2 or 89^2
Then the value of W:
W = 43 or 44
Now you can easily work out the rest.