The captain is twice as old as the first mate was when the captain was as old as the first mate is now.
How old is each of them today?
💡 Hint
The age gap between them never changes. Use it to say how many years ago the captain was the mate’s current age, then write one equation.
Reveal answer
The captain is 56 and the first mate is 42.
Step-by-step solution
- Let the captain be C and the mate M today. The captain is older, and the gap between them is C − M, which never changes.
- The captain was as old as the mate is now exactly C − M years ago.
- At that moment the mate was M − (C − M) = 2M − C years old.
- The captain is twice that age today: C = 2(2M − C), so C = 4M − 2C, which gives 3C = 4M.
- Combine with C + M = 98. From 3C = 4M, C = 4M/3, so 4M/3 + M = 98, which gives 7M/3 = 98 and M = 42. Then C = 56.
- Check: 14 years ago the captain was 42, the mate’s age now. The mate was then 28, and 56 is exactly twice 28. The ages add to 98.
Why it fools people
The sentence stacks a past moment inside a comparison inside another comparison, and our working memory runs out of room. Listeners try to picture both people at once in two different years and end up mixing them. The fix is purely clerical: name the unknowns, write the time gap down, and let algebra carry the load your memory can’t. The constant age gap does the rest.
Did you get it?
About This Puzzle
On a fishing boat working out of a New England harbor, the crew passes the long hours with arguments, card games and the occasional brain teaser. The captain is fond of one in particular because it is about her and her first mate, and because it sounds impossible the first time you hear it. There is no trick wording in the sense of a pun or a hidden meaning. Every word means exactly what it says. The difficulty is that one sentence folds two different moments in time into a single breath, and most listeners lose track of which age belongs to which year. Untangle it and the arithmetic is gentle.
How to Think About It
Don’t try to hold the whole sentence in your head. Break it into pieces and give each piece a name. Let the captain be C and the first mate be M today. The phrase “when the captain was as old as the first mate is now” points to a specific moment in the past, a certain number of years ago. Write that number of years in terms of C and M. Then ask how old the mate was at that moment. Finally, translate “the captain is twice as old as that” into an equation. A small table with a “now” column and a “then” column keeps everything straight.
Common Traps
The most common slip is to assume the captain’s age “then” and the mate’s age “now” are unrelated, which leaves too many unknowns. Another is to treat “twice as old as the first mate was” as twice the mate’s current age, which quietly drops the past tense. Some solvers also forget that both people age by the same number of years, so the gap between them never changes. That constant gap is the key to the whole problem, and it is easy to overlook when the sentence keeps jumping between tenses.
Where This Puzzle Comes From
Age puzzles built on this “when he was as old as she is now” phrasing are a long-standing tradition in recreational mathematics and school algebra, and one famous version about two people named Mary and Ann has been repeated for generations. This puzzle is an original setting in that family. The technique of tracking a constant difference over time carries over to many real problems, from interest rates to inventory.
Try This Variation
A bookstore owner is now 45. She is three times as old as her clerk was when the owner was as old as the clerk is now. How old is the clerk today?
Show the answer to the variation
The clerk is 30. With owner O = 45 and clerk K, the clerk was 2K − 45 back then, so 45 = 3(2K − 45), giving 6K = 180 and K = 30. Check: 15 years ago the owner was 30, and the clerk was 15. Three times 15 is 45.