Direct proportion

Twice as many pieces, twice as many korunas

A roll costs 4 Kč. Two rolls 8 Kč. Five rolls 20 Kč. See the rule? How many times more pieces, that many times more korunas. That is called direct proportion.

Two quantities are directly proportional when the growth of one pulls the growth of the other at the same pace. You double one, the other doubles. You triple — it triples.

The sign: the quotient of both quantities is always the same. Price divided by the number of rolls is always 4. That steady number is called the constant of proportion — here it is the price of one roll.

Watch, not everything in the world is directly proportional! A person’s height is not directly proportional to age (at 30 you do not measure 3 metres). Today you will learn to spot direct proportion, write it in a table and use the “via one” trick: first find the value for one piece, then multiply.

Jonáš and Sofie buy school notebooks. The sign: 3 notebooks for 54 Kč. Jonáš needs 7 notebooks and tries by eye: “So a hundred korunas?” Sofie goes via one: “One notebook: 54 : 3 = 18 Kč. Seven notebooks: 7 × 18 = 126 Kč.” Jonáš whistles: “Off by 26 korunas.” On the way home he tries to catch Sofie: “And if 10 notebooks?” — “180.” — “And 100?” — “1 800. As long as the price per piece stays the same, you just multiply. That is all of direct proportion.”

How to spot and use direct proportion

The direct proportion test: ask “if I double one, does the other double?” Rolls and price: yes. Age and height: no. Time of a ride at a steady speed and distance travelled: yes.

A table. Direct proportion is clearly written as a table of pairs:

pieces: 1, 2, 3, 5, 10 — price: 4, 8, 12, 20, 40.

In the table it holds: the quotient (price : pieces) is the same in every column — here 4. That is the constant of proportion. Written as a formula: price = 4 × pieces. In general y = k · x.

Calculating via one:

Step 1: from the given pair calculate the value for one (divide). Step 2: multiply by the number you want.

Three kilos of apples cost 87 Kč, I want 5 kg: one kilo 87 : 3 = 29 Kč, five kilos 5 × 29 = 145 Kč.

The graph of direct proportion (you will meet it in a textbook) is a straight line through zero: zero pieces, zero korunas. When the price “jumps” or starts somewhere other than zero (say because of postage), it is not direct proportion.

Six litres of petrol cost 216 Kč. How much do 10 litres cost?

Step 1 — via one: one litre costs 216 : 6 = 36 Kč.

Step 2: ten litres: 10 × 36 = 360 Kč.

Result: 360 Kč. Proportion test: twice as much petrol = twice as much money? Yes, the price per litre does not change with the amount. Check with an estimate: 10 litres is almost double six, the price should be almost double — 216 × 2 = 432, the result a bit less. That checks out.

Fill a direct proportion table

Given: 4 tickets cost 480 Kč. Fill the price for 1, 3, 7 and 11 tickets.

Step 1: constant (price per piece): 480 : 4 = 120 Kč.

Step 2: I multiply — 1 pc: 120 Kč, 3 pc: 360 Kč, 7 pc: 840 Kč, 11 pc: 1 320 Kč.

Check: the quotient price : pieces is 120 everywhere. That checks out.

The table is xong with one division and four multiplications. The constant of proportion is the key that opens the whole row.

Is it direct proportion? Three situations

a) The number of identical chocolate bars and the total price. Twice as many bars = twice the price. Yes.

b) A dog’s age and its mass. A two-year-old dog does not weigh twice a one-year-old. No — it grows unevenly and then stops.

c) The number of pages read at a steady pace and reading time. Twice as many pages = twice as much time. Yes.

The lesson: proportion is not a property of numbers, but of a relationship. Always do the doubling test before you start multiplying.

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.

Calculate via one: price per piece, then multiply. Write only the resulting amount.

Practise the questions

On paper: tables and tests

For each question write the constant of proportion — the number that belongs to one.

1. Five copies cost 35 Kč. Make a table for 1, 2, 8 and 12 copies.

2. A car travels 170 km in 2 hours at a steady speed. How far does it go in 5 hours? What is the constant of proportion and what does it mean?

3. Decide with the doubling test: is it directly proportional a) the amount of flour and the number of buns from a recipe, b) the number of diggers and the time for a hole, c) the side of a square and its perimeter?

4. Invent your own example of direct proportion from your day and write a table with four pairs for it.

Now you 💪

1. I know the test: I double one — does the other double?
2. I find the constant of proportion and I know what it means in the question.
3. I calculate via one: divide to the unit, times to the number I want.

What to take from this lesson

• Direct proportion: how many times more of one, that many times more of the other.
• The quotient of the quantities is steady — the constant of proportion (the value for one).
• The via-one trick: divide, then times.
• The doubling test catches fake proportions (discounts, bundles).
• The graph of direct proportion is a straight line from zero.

© 2026 Ing. Martin Polak / AlgoRhino · Điều khoản sử dụng nội dung