The rule of three · Kode bersama Martin

A universal writing with arrows that solves direct and inverse proportion without thinking where you divide and where you multiply.

You know three numbers, you calculate the fourth

The last few lessons you counted proportions with common sense: via one, via a constant product. Today you get one writing that handles both and you never get lost in it: the rule of three.

It is called that because you know three members (numbers) and you look for the fourth. The writing looks like this: two rows, in each a pair of values that belong together, the unknown you mark x.

3 notebooks ……… 54 Kč 7 notebooks ……… x Kč

Then come arrows. Along each column you draw an arrow in the direction the values grow. And now the rule: direct proportion = arrows the same way, inverse = arrows against each other.

From the arrows you build the calculation. It sounds like a ritual, but that ritual saves you in a test where in a panic you do not know what to divide by. The arrows always know.

Tereza and Sofie train for a proportion test. Sofie on the question “5 workers, 18 days, how many days for 9 workers” panics: “Do I divide by five? By nine? Do I multiply?” Tereza shows her the writing with arrows: workers grow (arrow up), days must fall (arrow down). “Arrows against each other — inverse. So 5 × 18 = 90 and divided by 9 is 10 days.” Sofie tries the next question herself and for the first time does not hesitate. “It works even under stress,” she wonders. Tereza: “That is why we learn it. For calm questions your head is enough, the arrows are for the nervous ones.”

💡

Draw the arrow in the direction from the first row to the second according to whether the value grows or falls. About the second column decide by thinking (will x be bigger or smaller?), not by guessing.

The rule of three step by step

Step 1: the writing. Two rows. In each belongs a pair of values that go together. The same quantities under each other, the same units under each other. The unknown is x.

Step 2: arrows. By the column with numbers draw an arrow in the direction of growth (from smaller to bigger). By the column with x decide by thinking: will the result be bigger or smaller than the known value?

Step 3: the decision. Arrows the same way = direct proportion. Arrows against each other = inverse.

Step 4: the calculation. Direct: x = known value × (second number of the column / first number). In practice: via one. Inverse: x = constant product / new value.

A full example (direct): 3 notebooks … 54 Kč, 7 notebooks … x. Catatanbooks grow ↑, korunas will grow ↑. Matching arrows. x = 54 : 3 × 7 = 126 Kč.

A full example (inverse): 4 taps fill a pool in 9 hours, 6 taps in x. Taps grow ↑, hours will fall ↓. Against each other. x = 4 × 9 : 6 = 6 hours.

A check at the end, always: does the result make sense by direction? More notebooks → more korunas (126 > 54, that checks out). More taps → fewer hours (6 < 9, that checks out).

Eight rolls cost 32 Kč. How much do 15 rolls cost?

Writing: 8 rolls … 32 Kč 15 rolls … x Kč

Arrows: rolls grow ↑, the price will grow ↑. Matching = direct proportion.

Calculation via one: 32 : 8 = 4 Kč a piece, then 15 × 4 = 60.

Result: 60 Kč. Direction check: 15 pieces is more than 8, the price 60 is more than 32. That checks out.

Six combine harvesters harvest a field in 10 hours. How many hours do 4 harvesters need?

Writing: 6 harvesters … 10 hours 4 harvesters … x hours

Arrows: harvesters fall ↓, the work will take longer ↑. Against each other = inverse proportion.

Calculation via a constant product: 6 × 10 = 60 harvester-hours. Then 60 : 4 = 15.

Result: 15 hours. Check: fewer machines, more time (15 > 10). That checks out. The arrows decided for me — I did not have to remember where you divide.

A units trap: 250 g of ham costs 45 Kč. How much does 1.2 kg cost?

First I match the units: 1.2 kg = 1 200 g.

Writing: 250 g … 45 Kč 1 200 g … x Kč

Matching arrows (more grams, more korunas) = direct.

Calculation: 45 : 250 = 0.18 Kč per gram. Then 1 200 × 0.18 = 216.

Result: 216 Kč. Who leaves kilograms next to grams in the writing counts nonsense 45 : 250 × 1.2. Units under each other must be the same — that is part of step 1, not a detail.

⚠️

The rule of three without deciding the arrows is guessing. The most common mistake: using the direct proportion method on an inverse question (more workers → “more days”). Draw the arrows and decide always, even if the question looks clear.

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 by the rule of three: via one. Write only the resulting amount.

Practise the questions

On paper: the arrows decide

Write each question in two rows, draw arrows, write the kind of proportion and only then calculate.

Four metres of fabric cost 260 Kč. How much do 7 metres cost?
Twelve workers put up scaffolding in 6 days. In how many days will 8 workers put it up?
A train goes at 80 km/h and covers the route in 3 hours. How long will it go after a track repair at 120 km/h?
Three copiers print flyers in 40 minutes. How many copiers do you need to be selesai in 24 minutes? (Watch: this time you look for the number of machines.)

Now you 💪

I can do a two-row writing with the same quantities and units under each other.
With arrows I decide the kind of proportion before I calculate.
At the end I check the result by direction: should it be bigger or smaller?

✅

Done when: Four questions have writing, arrows, kind of proportion and a check; the last one came out at 5 copiers.

What to take from this lesson

The rule of three: two rows, three known numbers, x.
Arrows the same way = direct, against each other = inverse.
Direct is solved via one, inverse via a constant product.
Units in a column must be the same — convert before calculating.
Checking the direction of the result is part of the question, not a bonus.