Problems on Age

At a glance 10 formulas

1. 1Past and future ages$n \text{ years on: } x + n \qquad n \text{ years ago: } x - n$
2. 2The gap never changes$(A + n) - (B + n) = A - B$$\text{parent's age at the child's birth} = \text{the gap}$
3. 3Sum and difference$A = \dfrac{S + D}{2},\quad B = \dfrac{S - D}{2}$
4. 4Ages in a ratio$a : b \Rightarrow ak \text{ and } bk$$\dfrac{ak + n}{bk + n} = \dfrac{c}{d}$
5. 5Ratio now, ratio later$k = \dfrac{n(d - c)}{bc - ad}$$a : b \text{ now},\ c : d \text{ after } n \text{ years; ago: use } -n$
6. 6Two ratios, one gapScale both ratios so the gap has the same number of parts; the parts one person gains over the years between give the size of a part.
7. 7Times as old$B = \dfrac{n(p - 1)}{m - p}$$A = mB \text{ now},\ A + n = p(B + n)$
8. 8When I was your age'When A was B's age' is one gap ago and 'when B is A's age' is one gap ahead, so shift every age by the gap.
9. 9A group grows older$\text{total after } t \text{ years} = \text{total} + Nt$$\text{the average rises by } t$
10. 10Back-solve from optionsAges in a ratio a : b are multiples of the part, so test only the options that fit, and reject any root that makes an age negative.