“In five years, twice as old” sounds like a riddle until you write the ages down. understand → set up → solve → verify both ages at the same moment. The gap never changes; the ratio does — that is the whole trick.
Two ages are known now. In how many years will the first person be a given multiple of the second?
In short
Age problems are equations about time: everyone gains one year per year, so the gap between two ages never changes while the ratio does. The setup is the same — write each age now, add or subtract the same t, and set the multiple the story gives. VerifySolve builds that equation in stages and checks both ages at the same moment.
Every age problem is the same frame: today’s ages, the same t added or subtracted for both people, and the multiple the story claims. The solver’s Stage 2 is that frame, filled in.
Ages in t years
Future multiple
Past multiple
Sum and difference
Each note re-tests the answer where the story claims it: the two ages at the future (or past) moment, with the multiple recomputed. Run them live in the solver above.
Maya is 32 and her daughter is 8. In how many years will Maya be twice as old?
In 16 years (ages 48 and 24)
Verified: 48 = 2 × 24 ✓, and the gap is still 24 years, as it must be.
A father is 3 times as old as his son. In 12 years he will be twice as old. How old are they?
The father is 36 and the son is 12
Verified both relations: 3 × 12 = 36 now ✓, and 2 × 24 = 48 in 12 years ✓.
A man is 40 and his son is 16. How many years ago was the man 3 times as old?
4 years ago (ages 36 and 12)
Verified: 36 = 3 × 12 ✓, and both ages back then are positive.
The sum of two sisters’ ages is 52 and the difference is 12. How old is each?
32 and 20
Verified: 32 + 20 = 52 ✓ and 32 − 20 = 12 ✓ — both facts hold.
Both people age. Writing 32 + t = 2 × 8 forgets that the daughter ages too — the equation must be 32 + t = 2(8 + t). One-sided aging is the single most common age-problem error.
Ratios shrink toward 1 as people age: someone twice as old as a child today will not be twice as old in twenty years. If a story claims the same multiple now and later, the ages are not pinned down — and the solver says so.
A negative answer usually means the multiple happened on the other side of "now" — or before someone was born. A negative age is never the answer; it is a sign the story points the other way, and the solver refuses it with the reason.
Because everyone gains exactly one year per year. If Maya is 24 years older than her daughter today, she was 24 years older ten years ago and will be 24 years older in ten years. The gap is constant — it is the ratio between the ages that moves, shrinking toward 1 as both people get older.
Write each person’s age now, then add the same t to both for the future (or subtract t for the past), and set the multiple the story gives: A + t = k(B + t). The single most common error is adding the years to only one person — both age, always.
A negative time means the event sits on the other side of "now" from the one you asked about: a negative "in how many years" answer is really a "how many years ago" answer. The solver refuses the impossible version and tells you which direction the story actually points.
The algebra can produce them (7½ years, say), and real textbook problems are usually designed to come out whole. The solver shows exact fractions rather than rounding, so a fractional answer is honest about what the numbers say — and its verification still checks out.
Both ages are computed at the same moment — future or past — and the story’s multiple is re-tested there: the older age must equal exactly k times the younger age, and both ages must be positive. A setup that fails either check does not ship.
Yes — free, no sign-up, no app, with the full staged solution and both-ages verification included.
The other classic "t years from now" family — motion instead of ages.
Two-equation stories: percents and prices, in stages.
The staged method explained, plus every word-problem family we cover.
Age clues that become two equations — solve the system directly.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.