Petes Age

Here is a problem by J.A.H. Hunter.

'Can't you find something better to do?' Pete's mother asked, busy fixing the guest room. 'I get the jitters with you following me around.' The boy grinned. 'I like watching you get ready for grandpa', he told her. 'How old is he?' His mother smiled. 'Why,' she replied. 'He's exactly your age and your cousin Ted's age multiplied together.' Pete considered the matter a moment. 'Ted's older than me,' he said, 'but I don't know how old he is.' 'Run away and figure it out,' his mother told him. 'The three ages add up to ninety years.' Of course you'll have to find out how old Pete was!

Please give all three ages and the method toget them.

Peter Heichelheim

spoiler.. . . . . . . . . . . . . . . . . . . . . . . . .

Let the three ages be a, b, and c where ab = c. Then a+b+c=90 a + b + ab = 90 (a+1)(b+1) = 91 Assuming the ages are whole numbers, the only factors of 91 are 1 and 91 or 7 and 13. Neither a nor b can be 0 so a = 6 and b = 12. Therefore Pete is 6, Ted is 12 and the grandpa is 72.

I got the same answer as you but J.A.H. Hunter gave Pete's age as 7. Anyone know what he was thinking about.

I'd guess he forgot that he solved for 'a+1', and not 'a' when factorizing 91. Even J.A.H. Hunter can make mistakes.

By the way, I got 6 and 12 as well: pt = g g + p + t = 90 pt + p + t = 90 pt + p + t + 1 = 91 p(t+1) + 1(t+1) = 91 (p+1)(t+1) = 91

since p<t, p+1=1 or 7 t+1=91 or 13

p = 0 or 6 t = 90 or 12 g = 0 or 72

I very much doubt that Pete's mom would be talking to Pete if a) Pete weren't born, or b) her father weren't born. Therefore, their ages are 6, 12, and 72 for Pete, Ted, and Grandpa

BASIC program:

* Variables: * p=Pete's age * g=Grampa's age * t=Ted's age * * data: * Ted is older than Pete * t * p = g * t + p + g = 90

integer p,t,g

for p=1 to 90 for t=1 to 90 for g=1 to 90 if t*p=g then if t+p+g=90 then if t>p then print 'got it! pete =';p;' ted =';t;'grampa =';g stop endif endif endif next g next t next p end

Program output:

got it! pete = 6 ted = 12 grampa = 72

Was that cheating?

let p=pete's age, t=ted's age. given: p + t + pt = 90 p(t+1) + t + 1 = 91 (add 1 to both sides) (p+1)(t+1) = 91 = 7 * 13 (primes) => p = 6, t = 12. grandpa's age is 6*12 = 72