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Age problems, demystified

By Sara Lin · Last updated April 2026 · 9 min read

The classic age problem looks like this: “Alice is three times as old as Bob. In ten years, Alice will be twice as old as Bob. How old are they now?” Algebraically the problem is trivial — two equations, two unknowns — but a remarkable number of students get stuck on the very first step, which is translating the English sentences into algebra.

This article spells out the translation rules and walks through the standard template. The calculator at the end handles the most common variant; once you have done the translation, the algebra itself is already covered in systems of linear equations.

The two-step recipe

Step 1 — name the unknowns

Pick the simplest unknowns possible — usually the current ages. Write them down explicitly. For our example:

“Let $A$ be Alice’s current age (in years), and $B$ be Bob’s current age (in years).”

Writing this down with a pen, in words, is not a formality. The hardest age problems are the ones where you accidentally let one variable mean “current age” and another mean “age in N years,” and then mix them up halfway through.

Step 2 — one equation per English sentence

“Alice is three times as old as Bob” becomes $A = 3B$. “In ten years, Alice will be twice as old as Bob” becomes $A + 10 = 2(B + 10)$. The key here is that “in ten years” means both their ages go up by ten, not just Alice’s. Forgetting to add the ten to Bob’s side is the single most common error in these problems.

With the system written out, the rest is just algebra. Substitute $A = 3B$ into the second equation:

$$3B + 10 = 2(B + 10) = 2B + 20.$$ $$3B - 2B = 20 - 10$$ $$B = 10.$$

And $A = 3B = 30$. Alice is $30$, Bob is $10$.

Common phrasings and how to translate them

The mistakes I see most often

1. Adding $n$ to only one person

“In ten years Alice will be twice as old as Bob” means both of them are ten years older. Writing $A + 10 = 2B$ instead of $A + 10 = 2(B + 10)$ is wrong, and the resulting answer will be slightly off — just enough to feel right but actually be incorrect. Always update both sides.

2. Mixing up “X times as old” with “X years older”

“Alice is three times as old as Bob” ($A = 3B$) and “Alice is three years older than Bob” ($A = B + 3$) are completely different equations. The first is multiplicative, the second additive. Misreading one for the other is the second most common mistake on age problems.

3. Negative or fractional ages

If your algebra produces $B = -5$ or $B = 7.4$, the puzzle is either ill-posed or you have set up an equation incorrectly. People do not have negative ages, and almost all textbook problems are designed to have integer answers.

A few harder variants

“The sum of two siblings’ ages is $30$. In five years one will be twice as old as the other. How old are they now?”

Let $A$ and $B$ be the current ages. The two equations are $A + B = 30$ and $A + 5 = 2(B + 5)$. From the first, $A = 30 - B$. Substitute into the second: $30 - B + 5 = 2B + 10$, so $25 = 3B$, giving $B = 25/3$ — which is not a whole number. Either the question expects a fractional answer (unlikely) or there is a typo in the problem (likely). Always sanity-check.

“Three years ago, the father was four times as old as the son. In seven years, he will be twice as old. Find their current ages.”

Let $F$ and $S$ be current ages. “Three years ago” gives $F - 3 = 4(S - 3)$. “In seven years” gives $F + 7 = 2(S + 7)$. Expand: $F = 4S - 9$ and $F = 2S + 7$. Set them equal: $4S - 9 = 2S + 7$, so $S = 8$ and $F = 23$. Always check both original conditions: three years ago, $F - 3 = 20$ and $4(S - 3) = 20$ — matches. In seven years, $F + 7 = 30$ and $2(S + 7) = 30$ — matches.

Where this comes up later

Age problems are not really about age; they are practice for the broader skill of translating English into algebra, which underpins every word problem you will ever see. The same setup discipline applies to mixture problems (“how much $X\%$ solution should you add to a $Y\%$ solution to get $Z\%$?”), work-rate problems (“A can finish in 4 hours, B in 6, how long together?”), and most physics word problems.

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