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11+ Maths Guide · Year 5–6 · Free lesson

Algebra Problem Solving

The final algebra lesson chains the whole topic: turn 'I think of a number' stories into equations, solve them, and crack the classic sum-and-difference puzzle — always checking against the story itself.

A12Algebra Problem Solving
Puzzles with unknowns, exam style.
Combine everything: translate the puzzle, solve the equation, check the answer.
‘I think of a number, double it and add 5. I get 17.’ What was the number?
2n + 5 = 17
− 5 → 2n = 12  ·  ÷ 2 → n = 6
Check: double 6 is 12, add 5 is 17 ✓
Translate → solve → check. Every puzzle, same three beats.
1

Learn the 3-Step Method

1Translate the words.

‘Double it and add 5, get 17’ → 2n + 5 = 17.

words → 2n + 5 = 17

2Solve with inverses.

−5 then ÷2: n = 6.

n = 6

3Check against the story.

Run the story with 6: double → 12, add 5 → 17 ✓.

story check ✓
💡 Key IdeaTwo unknowns? Use both facts together: a + b = 10 and a − b = 4 → a = 7, b = 3 (their sum and difference pin them down).
2

Worked Example

Two numbers add to 10; their difference is 4. Find them.

Add the facts
sum 10, difference 4:
the bigger = (10 + 4) ÷ 2
14 ÷ 2 = 7
→
Find the smaller
10 − 7 = 3
→
✓ Check
7 + 3 = 10 ✓ · 7 − 3 = 4 ✓
7 and 3
3

Watch Out! Common Mistakes

✕
Translating in the wrong order.
“double it and add 5” → 2 × (n + 5) ✕

Follow the story’s order! Double FIRST, then add: 2n + 5. (The bracketed version doubles the 5 too.)

✕
Halving the sum for a sum-and-difference puzzle.
10 ÷ 2 = 5 and 5 ✕

Their difference is 4! 5 and 5 differ by 0. Use (10+4)÷2 = 7 and 10 − 7 = 3.

✕
Checking against your equation, not the story.

Check the STORY! If you translated wrongly, your answer can fit your (wrong) equation perfectly. Re-run the original words.

⬇ Cheat Sheet

ALGEBRA PROBLEM SOLVING
  1. Translate the story in ITS order into an equation.
  2. Solve with inverse operations.
  3. Two unknowns: bigger = (sum + difference) ÷ 2.
  4. Check against the original story, not your equation.
Examples: 2n + 5 = 17 → n = 6 · sum 10, diff 4 → 7 and 3.

📖 Continue Your Journey

✓ A1 What Is Algebra?✓ A2 Function Machines✓ A3 Writing Expressions✓ A4 Substitution✓ A5 Simplifying Expressions✓ A6 One-Step Equations✓ A7 Two-Step Equations✓ A8 Sequences✓ A9 The nth Term✓ A10 Formulae✓ A11 Bar Models for UnknownsA12 Algebra Problem Solving
→
You’ve completed the Algebra Journey! 🎉Back to the Algebra Hub
📄 Free printable: Download the Algebra Problem Solving cheat sheet (PDF) — keep it by your child's desk.

Quick summary

Frequently asked questions

How do you solve 'I think of a number' puzzles?

Translate the story into an equation in the same order (double it, add 5, get 17 → 2n + 5 = 17), solve with inverses, then re-run the story with your answer as a check.

How do you find two numbers from their sum and difference?

The bigger is (sum + difference) ÷ 2; the smaller is the sum minus that. For sum 10 and difference 4: 7 and 3.

Why check against the story rather than the equation?

Because a mistranslated equation can be solved perfectly and still give the wrong answer to the actual question. The story is the source of truth.

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