# How old?

How old are these three people?

## Problem

Joseph recently noticed that this time next year he will be twice as old as Jasmine.

I couldn't help noticing that when I am 66 at this time of year the combined future ages of Joseph and Jasmine will also be 66.

Also, 6 years from now my future age will be a multiple of the sum of Joseph and Jasmine's future ages. Of course, all ages are non-negative integers.

How old are Joseph, Jasmine and I now?

I couldn't help noticing that when I am 66 at this time of year the combined future ages of Joseph and Jasmine will also be 66.

Also, 6 years from now my future age will be a multiple of the sum of Joseph and Jasmine's future ages. Of course, all ages are non-negative integers.

How old are Joseph, Jasmine and I now?

## Getting Started

## Student Solutions

Thanks to Thomas Udale from Winchester College for this solution, which was the age of me and my two children at the time of writing the problem :-)

Joseph's age = x

Jasmine's age = y

your age = z

from sentence 1 : x+1=2(y+1)

from sentence 2: 66 = x+y+2n = z+n

from sentence 3: z+6=m(x+y+12)

z < 66

z+6 < 72

m(x+y+12) < 72

m(x+y+12) = m(3y+13)

m must be a positive integer, because it is the multiple, greater than 1

if m = 4, then m(3y+13) > 72

if m = 2:

then z+6 = 6y+26

from equation 1 into equation 2 : z + n = 3y +1 +2n

z = 6y+ 20 = 3y + 1 + n

n = 3y + 19

equation 2:

3y + 2(3y+12) +1 = 66

y = 3,

x = 7,

z = 38,

if m = 3,

following same as above,

n = 6y + 38

3y + 2(6y+38) + 1 = 66

15y + 77 = 66,

so y would be negative.

Joseph's age = x

Jasmine's age = y

your age = z

from sentence 1 : x+1=2(y+1)

from sentence 2: 66 = x+y+2n = z+n

from sentence 3: z+6=m(x+y+12)

z < 66

z+6 < 72

m(x+y+12) < 72

m(x+y+12) = m(3y+13)

m must be a positive integer, because it is the multiple, greater than 1

if m = 4, then m(3y+13) > 72

if m = 2:

then z+6 = 6y+26

from equation 1 into equation 2 : z + n = 3y +1 +2n

z = 6y+ 20 = 3y + 1 + n

n = 3y + 19

equation 2:

3y + 2(3y+12) +1 = 66

y = 3,

x = 7,

z = 38,

if m = 3,

following same as above,

n = 6y + 38

3y + 2(6y+38) + 1 = 66

15y + 77 = 66,

so y would be negative.