Word problems on multiplying and dividing · Lập trình với Martin

How to tell in a problem when to multiply and when to divide, and how to write a write-up with an answer.

Times, or divided by? The problem will tell you

You already know how to calculate. A word problem adds one extra step: decide what to calculate. And that step is what makes a reader a mathematician.

The key question sounds: do I put piles together, or split them?

I put equal piles together → I multiply. “They bought 4 packs of 6 cards” — four piles of six, 4 × 6.

I split a whole into equal piles → I divide. “30 sweets they shared among 5 children” — a whole into piles, 30 ÷ 5.

Signal words help: of, each, a box/pack/row point to multiplying. Share, fairly, among people, how many in each point to dividing. But watch — words only hint. What decides is the picture: close your eyes and see whether piles grow together, or a whole breaks into pieces.

And every problem has three required parts: a write-up, a calculation, an answer in a full sentence. Without an answer the calculation is only a number with no address.

Jonáš got two problems as homework. First: “In the class there are 6 desks of 4 notebooks. How many notebooks?” Second: “Share 24 notebooks evenly among 6 desks. How many on each?” Jonáš complains to mum that they are the same — all sixes and fours and twenty-fours. Mum advises drawing: for the first he draws desks and on each 4 notebooks — piles grow, 6 × 4 = 24. For the second he has a pile of 24 notebooks and shares — the whole breaks, 24 ÷ 6 = 4. Same numbers, opposite story. From then on Jonáš scribbles a picture for every problem.

💡

If you do not know whether times or divided by, draw the problem: circles as piles, marks as things. The picture will decide for you — growing piles is times, breaking a whole is divided by.

Method for every word problem

Five steps, the same every time:

1. Read twice. First for the story, second for the numbers. Do not rush — half the mistakes in word problems come from a bad reading.

2. Make a write-up: list what you know and what you look for:

packs … 4 in a pack … 6 cards altogether … ? cards

3. Decide the operation. Do I put piles together (times), or break a whole (divided by)? Draw a picture if you like.

4. Calculate — and with big numbers estimate first.

5. Answer in a full sentence and check the sense: “Altogether they have 24 cards.” Does the answer fit the question? Is it suspiciously big or small?

Two kinds of division you will meet:

Sharing: “30 sweets for 5 children. How many does each get?” → 30 ÷ 5 = 6 sweets.
How many in each: “We pack 30 sweets in 5s. How many bags do we fill?” → also 30 ÷ 5 = 6, but bags!

Same calculation, different answer — that is why you answer in a full sentence. The sentence forces you to say what there are six of.

A multiplying problem

“Ema bought 4 packs of stickers of 6 pieces. How many stickers does she have?”

Write-up: packs … 4, in a pack … 6, altogether … ?

Decision: I put piles (packs) together → times.

Calculation: 4 × 6 = 24.

Answer: Ema has 24 stickers altogether.

Sense check: more than one pack → more than 6. It fits.

A division-with-remainder problem

“17 children go on a trip. 5 children fit in one car. How many cars are needed?”

Write-up: children … 17, in a car … 5, cars … ?

Calculation: 17 ÷ 5 = 3, remainder 2.

Watch the answer! Three cars take only 15 children — two would stay on the pavement. The remainder here means: I need another car.

Answer: 4 cars are needed.

With a remainder always think through what it means in the story. Sometimes you round up (cars), other times the remainder simply stays (sweets).

A two-step problem

“Tereza bought 3 packs of yoghurts of 4 pieces and 2 extra yoghurts. How many yoghurts does she have?”

Write-up: packs … 3 of 4, extra … 2, altogether … ?

Step 1 (piles): 3 × 4 = 12 yoghurts in packs.

Step 2 (add the loose ones): 12 + 2 = 14.

Answer: Tereza has 14 yoghurts.

In one write-up: 3 × 4 + 2 = 14 — and the order of operations from the tenth lesson makes sure you multiply first. Word problems are the place where all the lessons meet.

⚠️

Do not hunt numbers and add them blindly! The problem “4 packs of 6” is not 4 + 6 = 10. Whoever calculates without reading the story hits the operation only by luck. Read twice, draw, then calculate.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Bắt đầu with ten. When it goes well, add more.

Warm-up: times tables forwards and backwards, so you do not hunt them in word problems.

Practise the questions

On paper

Each problem: a write-up, a picture (small is fine), a calculation, an answer in a sentence.

Solve: “In the cinema there are 8 rows of 9 seats. How many people fit?”
Solve: “Share 56 marbles fairly among 7 children. How many does each get?”
Solve with thought about the remainder: “We put 39 eggs into cartons of 6. How many full cartons do we fill and how many eggs are left?”
Make up your own word problem in which 5 × 8 comes out, and a second in which 40 ÷ 8 comes out. Give them to someone at home to solve.

Now you 💪

You know five steps: read twice, write-up, decide the operation, calculate, answer in a sentence.
You tell putting piles together (times) from breaking a whole (divided by) even with the same numbers.
With division with a remainder you think through what the remainder means for the answer.

✅

Done when: You take a word problem apart into a write-up, you correctly choose times or divided by and you answer in a full sentence that makes sense.

What to take from this lesson

I put equal piles together → I multiply. I break a whole → I divide.
Signal words (of, each, share) hint, the picture decides.
Required trio: write-up, calculation, answer in a full sentence.
Think through a remainder in a problem: sometimes it means an extra car, other times leftover sweets.
Two-step problems join times with plus — the order of operations keeps order.