Skill: word-problems

Multi-Step Word Problems

Word problems are not arithmetic in disguise — they are detective work. A four-step strategy (understand, plan, solve, check) turns any multi-step story into a sequence of small, safe computations.

1 Understand

The core idea in plain language.

Step 1 — Understand: read the whole story once without touching a pencil. Then underline the question — the exact thing being asked. Circle the numbers that matter and cross out the decoys. If you cannot say what the answer will look like (“an amount of money left over”), you are not ready.

Step 2 — Plan: work backwards from the question. “To find the change, I need the total cost and the amount paid.” Each “I need” becomes one computation step. Write the steps as a numbered list before you compute anything.

Step 3 — Solve: compute one step at a time, labeling each intermediate result (“total cost = $13.50”). Never do two operations in your head at once — that is where most errors hide.

Step 4 — Check: re-read the question. Did you answer what was asked — not just step one? Then sanity-check against the story: change must be less than what you paid; a total must exceed each part.

2 See it

Diagrams that make the idea visual.

The four-step attack plan

Every multi-step problem bows to the same routine:

Understand — underline the question, circle useful numbers.
Plan — list the computations, in order.
Solve — one step at a time, label results.
Check — answer the actual question; sanity-check the size.

Understand → Plan → Solve → Check. Skipping a step is how points are lost.

Bar model: see the story

Draw the quantities as bars. A total bar splits into part bars; a comparison puts two bars side by side so the difference jumps out.

If you can draw the bars and label them, the operations choose themselves: parts → add, missing part → subtract, equal groups → multiply or divide.

Comparison bars make “how many more” visible: 45 − 28 = 17.

3 The key rule

The check step

answer the question asked, then sanity-check the size

After computing, re-read the question word for word. Then ask: is my answer a sensible size for this story? Change < amount paid. A total > each part. A per-item price < the total.

4 Worked examples

Follow each step. The pattern is always the same.

Example 1 Two steps: total, then change

Understand: Maya buys 3 notebooks at $4.50 each and pays with $20. Question: her change.
Plan: (1) total cost = 3 × $4.50; (2) change = $20 − total.
Solve: 3 × $4.50 = $13.50; $20.00 − $13.50 = $6.50.
Check: answered “change” (not the total); $6.50 < $20. Sensible.

Maya gets $6.50 back.

Check: Reverse: $13.50 + $6.50 = $20.00. Correct.

Example 2 Comparison: how many more

Understand: Maya has 45 stickers, Leo has 28. Question: how many more does Maya have?
Plan: one subtraction: 45 − 28.
Solve: 45 − 28 = 17.
Check: 17 < 45, and 28 + 17 = 45. Correct.

Maya has 17 more stickers.

Check: Parts rejoin: 28 + 17 = 45. Correct.

Example 3 Three items, one total

Understand: 2 pencils at $3.25, 1 eraser at $5.75, 3 pens at $1.50. Question: total.
Plan: three multiplications, then add.
Solve: $6.50 + $5.75 + $4.50 = $16.75.

The total is $16.75.

Check: Estimate: $6 + $6 + $5 = $17 ≈ $16.75. Correct.

Example 4 Money left after spending

Understand: Start with $20, buy 4 folders at $3.75. Question: money left.
Plan: (1) spent = 4 × $3.75; (2) left = $20 − spent.
Solve: 4 × $3.75 = $15.00; $20 − $15 = $5.00.

$5.00 left.

Check: $15 + $5 = $20. Correct.

5 Common mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

1. Grabbing numbers and guessing the operation

Wrong

Seeing 45 and 28 and adding: “45 + 28 = 73 stickers.”

Right

Underline the question first: “how many more” → subtract: 17.

The question word picks the operation, not the numbers. Underline it before you compute.

2. Stopping after step one

Wrong

Computing the $13.50 total and circling it as the answer — but the question asked for change.

Right

Re-read the question after computing: change = $20 − $13.50 = $6.50.

Step one is rarely the answer. Re-read the question after every computation.

3. Skipping the sanity check

Wrong

“$65 change from a $20 bill” — written down without blinking.

Right

Change must be less than what you paid. $65 > $20 is impossible — recompute.

Every story has size rules. If the answer breaks them, the arithmetic is wrong.

6 Key vocabulary

Say these words like you mean them.

Multi-step problem: A story needing two or more computations to reach the answer.
Intermediate result: The answer to one step, used by the next step. Label it.
Decoy number: A number in the story you do not need. Cross it out.
Total: The combined amount of the parts.
Change: Amount paid minus total cost — always less than the amount paid.
Per item: The price of one item: total ÷ number of items.

7 Quick check

Try each one on paper first, then reveal the answer.

1. Ana buys 2 books at $6.25 each and pays with $15. What is her change?

Answer

Total $12.50; change $15 − $12.50 = $2.50.

2. Tom has 52 marbles, Sara has 37. How many more does Tom have?

Answer

52 − 37 = 15.

3. A pack of 6 yogurts costs $5.40. What is the price per yogurt?

Answer

$5.40 ÷ 6 = $0.90.

4. You start with $30 and buy 5 markers at $2.80 each. How much is left?

Answer

Spent $14.00; left $30 − $14 = $16.00.

Key points to remember

Understand → Plan → Solve → Check — in that order, every time.
Underline the question before you touch the numbers; the question word picks the operation.
Plan backwards from the question: each “I need” is one computation step.
Label every intermediate result (“total cost = $13.50”) — never chain two operations mentally.
Re-read the question after computing: did you answer what was asked?
Sanity-check the size: change < paid, total > each part, per-item < total.

Practice problems Try the tool Explore the lab