Practice · Grade 6 Foundations
Rule of Three — Practice Problems
Practice questions on the direct and inverse rule of three in rising difficulty. A hint and a full solution for every problem. Free, no signup.
Q1 of 7
0 correct
Direct: 3 kg cost 12 €. What do 5 kg cost?
3 kg → 12 €, 5 kg → ?
Quick answer
How do I best practise the rule of three?
Settle the direction first on every problem: if both quantities grow together it is direct — work through the value for one unit (scale down to 1, then up to the amount you want). If one rises while the other falls it is inverse — multiply the two known values into a product and divide by the new amount. Write the pairing as a → b, do a sanity check, and practise direct and inverse problems mixed so you reliably spot the direction.
HowTo
Solving strategy in 4 steps
This order fits every rule-of-three problem — whether it is direct or inverse proportion.
- 1Step 1 of 4
Write the known pairing
Note the relationship as a → b, e.g. 3 kg → 12 €. Keep careful track of which quantity is on the left (amount) and which is on the right (value). Mark the wanted quantity with a question mark.
- 2Step 2 of 4
Settle the direction of the proportion
Ask yourself: when one quantity grows, does the other grow with it (direct) or shrink (inverse)? More kilograms → more euros is direct. More workers → less time is inverse. This is the most important step.
- 3Step 3 of 4
Choose the right working
Direct: scale down to 1 (b ÷ a), then up to the new amount c (c · value-for-one). Inverse: form the product a · b and divide by the new amount c. Direct example: 12 ÷ 3 = 4, then 5 · 4 = 20.
- 4Step 4 of 4
Check the result
Sanity check: in a direct rule of three more amount must give more value (5 kg cost more than 3 kg). In an inverse one more amount gives less value (more workers, less time). If the size is off, you mixed up the direction.
Examples
Worked practice examples with full working
Typical questions from Grade 6–7 tests. Try each one yourself first, then compare with the solution.
Easy
Direct: 3 kg → 12 €, 5 kg → ?
Directly proportional: more kg, more €
1 kg → 12 ÷ 3 = 4 €
5 kg → 5 · 4 = 20 €
Check: 20 ÷ 5 = 4 € per kg = 12 ÷ 3 ✓
Basic direct rule of three: scale down to 1, then up to the amount you want.
Medium
Direct: 250 km on 5 L, 400 km → ?
Directly proportional: more km, more litres
1 km → 5 ÷ 250 = 0.02 L
400 km → 400 · 0.02 = 8 L
Check: 8 ÷ 400 = 0.02 L per km = 5 ÷ 250 ✓
The method is the same even with decimal values-for-one.
Medium
Inverse: 4 workers → 6 h, 3 workers → ?
Inversely proportional: fewer workers, more time
Product: 4 · 6 = 24 worker-hours
3 workers → 24 ÷ 3 = 8 h
Check: 3 · 8 = 24 = 4 · 6 ✓
Inverse rule of three: form the product, then divide by the new amount.
Hard
Boss: 5 painters → 7.2 h, 9 painters → ?
More painters → less time, so inverse
Product: 5 · 7.2 = 36 painter-hours
9 painters → 36 ÷ 9 = 4 h
Check: 9 · 4 = 36 = 5 · 7.2 ✓
Spot the direction first, then use the constant product. Here with a decimal.
Pitfalls
Common mistakes — and how to avoid them
These five traps show up in almost every rule-of-three problem.
Mixing up the direction of the proportion
“More workers need more time” is wrong — more workers need less time. Always settle first whether the other quantity grows along (direct) or shrinks (inverse).
Multiplying instead of dividing in the direct case
Scaling down to 1 is a division (12 ÷ 3 = 4), not a multiplication. Divide first, then multiply by the new amount.
Treating an inverse problem like a direct one
In an inverse rule of three you do not scale up through a value-for-one; you form the product a · b and divide by the new amount c.
Swapping amount and value
Amount belongs on the left, value on the right — consistently. Mixing the columns flips the fraction and gives the reciprocal of the right answer.
Misplacing the decimal point
With values-for-one like 0.02 L per km every decimal place matters. Write intermediate results exactly and only round at the very end.
Study
Practise with a plan — three short tips
Mix direct and inverse problems
Practising only direct problems back to back means you miss the inverse ones in a test. Mix the types on purpose so you check the direction afresh each time.
Solve first, then look at the answer
Write your working before you reveal the hint. Active recall is three to four times more effective for learning than passive reading.
On every wrong answer, ask why
Wrong direction? Multiplied instead of divided? Columns swapped? Note the cause — next time you will recognise the mistake right away.
FAQ
Frequently asked questions about practising
Glossary
Terms in one sentence
- Rule of three
- A method to find the fourth value of a proportion from three known values.
- Directly proportional
- More means more: both quantities grow in the same ratio.
- Inversely proportional
- More means less: as one quantity rises the other falls; their product stays constant.
- Value for one
- The value of a single unit (b ÷ a), the intermediate step in the direct rule of three.
- Proportion
- Equality of two ratios, e.g. 3 : 12 = 5 : 20.
- Ratio
- The relationship between two quantities, written a : b.
- Boss question
- The last and hardest problem in a practice set, combining several traps.