Two kinds of connection

Liri had to buy fuel – and later also share out the supplies for her crew. Both involved numbers that are connected. But in completely different ways.

Erklärus: “First recognise the kind: if both quantities rise together, it is direct proportion. If one rises while the other falls, it is inverse. The rest is logic.”

🟡 yellow = multiplied  ·  🔵 blue = divided  ·  🟢 green = the result

⛽ Direct proportion – the more, the more

Liri needs 15 units of fuel. 10 units cost 240 credits. The trick: first work down to one unit, then scale up to the amount you want.

FuelPriceStep
10240known
124÷ 10
15360× 15

Step 1: 240 ÷ 10 = 24 (unit value). Step 2: 24 × 15 = 360 (the amount you want).

🔵 first divide (down to 1) → 🟡 then multiply (up to the amount) → 🟢 result

Liri: “So always divide by the known amount first, then multiply by the amount I want?”
Erklärus: “Exactly. And if you forget the method: 15 is 1.5 times as much as 10, so it also costs 1.5 times as much – 240 × 1.5 = 360. Same answer, different logic.”
8 biscuits cost 40 cents. What do 3 biscuits cost?

🍱 Inverse proportion – the more, the less

The supplies last for 4 people, 12 days. But 2 more passengers come on board. How long do the supplies last now for 6 people?

Liri: “More people – fewer days. That is the other way round!”
Erklärus: “Exactly – inverse proportion. Here it is not the unit value that stays the same, but the PRODUCT of the two quantities.”
PeopleDaysStep
4124 × 12 = 48
648total supply
6848 ÷ 6

Step 1: total supply = 4 × 12 = 48 person-days. Step 2: 48 ÷ 6 = 8 days.

🟡 first multiply (find the total supply) → 🔵 then divide (by the new number) → 🟢 result

3 machines need 8 hours. How long do 4 machines need for the same job?
🌌 The unitary method in the universe

Fuel consumption is usually in direct proportion to the distance: double the distance, double the consumption.
How long a supply lasts is in inverse proportion to the size of the crew: more crew, fewer days.
Space engineers have to recognise first, in every calculation, what kind of connection there is.

❌ Common mistake

The most common mistake: mixing up direct and inverse proportion – then, with “more people”, you wrongly calculate with more days as well!

Tip: first ask yourself: if one quantity rises, does the other one rise too (direct) or fall (inverse)? Only then calculate.

⛽ Exercise 1 – Direct
5 star maps cost €25. What do 8 star maps cost?
🍱 Exercise 2 – Inverse
Supplies last 3 people 20 days. How many days do they last for 5 people?
🤔 Exercise 3 – Recognising
“6 workers build a spaceship in 10 days.” Is that in direct or inverse proportion to the number of workers?
direct (more workers → more days)
inverse (more workers → fewer days)

Two ways, one principle

Liri had worked out her fuel and redistributed the supplies – two quite different calculations, but both with clear logic instead of formulas learned by heart.

Liri: “I don’t have to remember a formula – I only have to understand whether the two quantities grow together or in opposite directions.”
“Direct proportion: the unit value stays the same (÷ then ×). Inverse proportion: the product stays the same (× then ÷). First recognise the kind, then calculate.”
What you learned today
Direct: ÷ then ×  ·  Inverse: × then ÷
First recognise: do both grow together or in opposite directions?
🌟

Chapter 40 mastered!

Liri can now tell the two kinds of the unitary method apart and use them.

Erklärus’s notebook – page 40:
“Unitary method: first recognise the kind (direct = both grow together, inverse = one grows, the other falls). Direct: down to 1, then up (÷ then ×). Inverse: the product stays constant (× then ÷).”
← Back to overview 40 – Unitary Method