Word problems with the rule of three

A mix of questions on direct and inverse proportion: recipes, building, speeds. The arrows decide.

The arrows decide even where words confuse you

You know the rule of three. Today we let it out into the wild — onto word problems where nobody announces “this is direct proportion”. You have to spot it yourself.

The key question stays: more means more, or more means less? More bakers bake more rolls (direct). More bakers finish a job in fewer hours (inverse). The same bakers, a different question — so decide again for every question.

The method for every question: a two-row writing, arrows, kind of proportion, calculation, direction check. Five steps, no rush.

And one extra novelty: questions with a unit conversion. Minutes vs. hours, grams vs. kilograms. The rule of three works only when the same quantities under each other are in the same units — converting is step zero, still before the writing. Today the questions will be more mixed, but you have the tool complete in your hand.

Ema and Tereza count supplies for a class weekend: 15 children use 6 litres of drink a day. “And 20 of us are going and we will be there 2 days,” says Tereza. Ema solves in steps: “Children first: 15 children … 6 litres, 20 children … x. More children, more drink, direct: 6 : 15 × 20 = 8 litres a day.” Tereza: “And two days — twice as much. 16 litres.” Ema shakes her head: “Two rules of three one after another. One for children, the other for days. Do not try to swallow it all at once.”

💡

Split a hard question into two rules of three one after another — first one change, then the other. Two simple rules of three are more reliable than one awkward one.

The rule of three in the wild

Step 0: units. Read the question and match the units: everything in minutes, or everything in hours. 1.5 h = 90 min; 2.4 kg = 2 400 g.

Step 1: writing pairs. What belongs with what? 5 metres of fabric … 320 Kč. Not amounts to days when they belong to metres.

Step 2: arrows and a decision. Will x grow or fall? Decide by thinking about the world, not by looking at the numbers: more workers build faster, a longer route takes longer, a stronger engine pulls more.

Step 3: the calculation. Direct: via one. Inverse: a constant product.

Step 4: a check of direction and sense. Answer in a full sentence and ask: does this make sense? A cyclist will not go 200 km/h, a snack will not cost 40 000 Kč.

A double question (two quantities change): solve in order. First fix one change, calculate a middle result, then apply the second. Like in the scene: first more children (direct), then more days (direct). Direct can mix with inverse too — that is why you decide at each step separately.

Three metres of fencing cost 255 Kč. How much does a 14 m fence cost?

Step 0: the units fit (metres and korunas).

Writing: 3 m … 255 Kč / 14 m … x Kč. More metres, more korunas — arrows matching, direct.

Calculation via one: 255 : 3 = 85 Kč a metre. Then 14 × 85 = 1 190.

Check: 14 m is almost five times 3 m, the price almost five times 255 (≈ 1 275). The result 1 190 is close. That checks out.

Answer: the fence will cost 1 190 Kč.

Six pumps fill a tank in 90 minutes. Will 9 pumps manage it in an hour?

Step 0: an hour = 60 minutes — converted, I compare minutes with minutes.

Writing: 6 pumps … 90 min / 9 pumps … x min. More pumps, less time — arrows against each other, inverse.

Calculation: constant product 6 × 90 = 540. Then 540 : 9 = 60 minutes.

Answer: 9 pumps fill the tank in exactly 60 minutes — they manage it. Direction check: pumps one and a half times more, time one and a half times shorter (90 → 60). That checks out.

A double: 4 cooks make 120 portions in 3 hours. How many portions will 6 cooks make in 2 hours?

I solve twice, I always change only one thing.

Change 1 — cooks (hours fixed at 3): more cooks, more portions, direct. 120 : 4 × 6 = 180 portions in 3 hours.

Change 2 — time (cooks fixed at 6): fewer hours, fewer portions, direct. 180 : 3 × 2 = 120 portions.

Answer: 6 cooks in 2 hours make 120 portions — the same as the original crew, because more hands exactly balanced the shorter time (6 × 2 = 12 = 4 × 3 cook-hours).

⚠️

Do not decide the kind of proportion by the numbers, but by the world. The numbers 4 and 6 look the same in a question about prices and about workers, but once you multiply and once you divide. What decides is the sentence “more means more/less” about the real situation — write it every time.

Practice: now you do it 🎲

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

Direct proportion in questions about prices. Write only the resulting amount.

Practise the questions

On paper: questions from the wild

Five steps on each: units, writing, arrows, calculation, answer in a full sentence.

For 8 pancakes you need 250 g of flour. How much flour for 20 pancakes?
A stock of feed lasts 12 rabbits 15 days. How many days will it last when you buy 6 more rabbits?
A car uses 6.5 litres per 100 km. How many litres does it need for a route of 240 km? (Watch the writing of pairs.)
A double: 3 bricklayers build a wall in 8 days. In how many days will 4 bricklayers build two such walls?

Now you 💪

Before writing I match the units and the pairs that belong together.
I decide the kind of proportion with a sentence about the real world, I confirm it with arrows.
I solve double questions one change at a time and I answer in a full sentence.

✅

Done when: Four questions solved in five steps: 625 g, 10 days, 15.6 litres and 12 days — all with an answer in a full sentence.

What to take from this lesson

Step zero: match the units, pair the right pairs.
The kind of proportion is decided by the world, not the numbers.
Two changes = two rules of three one after another.
A sense check: the result must make sense in the story.
An answer in a full sentence closes the question.