KA-0111Age problems
4 points, difficulty 2 of 3Level 5–6about 90sA mother is 32 and her child is 8. In how many years will the mother be exactly three times as old as the child?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Age both of them, not just one.
2The strategy
In x years the mother is 32 + x and the child is 8 + x. Write the condition as an equation between those two and solve it. The difference between their ages never changes.
3The full solution
32 + x = 3(8 + x) gives 32 + x = 24 + 3x, so 8 = 2x and x = 4. Check: 36 is three times 12.
Solution
The reliable way
In x years the mother is 32 + x and the child is 8 + x. The condition is 32 + x = 3(8 + x), which is 32 + x = 24 + 3x, so 8 = 2x and x = 4. Check: in 4 years they are 36 and 12, and 36 is exactly three times 12. The transferable idea: everyone ages at the same rate, so add the same x to every age - and the difference between two ages never changes.
The elegant way
The gap is always 24. When the mother is three times as old, the gap is two of the child's ages, so the child is 12 - which is 4 years away.
Why this is on the test: Ageing only one person is the classic mistake, and the constant difference is the fact that makes the elegant route possible.