Word problems with ratios and the rule of three
Ratios and the rule of three are two of the workhorses of grade 7. A ratio tells you "in what proportion" something is split; the rule of three answers "how much do I get for a different amount".
Two-part ratio
Problem. Mia and Jay share £60 in the ratio 2 : 3. How much more does the bigger share get?
Working:
- Sum of the parts: `2 + 3 = 5` parts.
- One part is worth: `60 / 5 = £12`.
- Smaller share: `2 · 12 = £24`. Bigger share: `3 · 12 = £36`.
- Difference: `36 − 24 = £12`.
Three-part ratio
Problem. A profit of £90 is split in the ratio 1 : 2 : 3. How much is the middle share?
Working:
- Sum: `1 + 2 + 3 = 6` parts.
- One part: `90 / 6 = £15`.
- Middle share: `2 · 15 = £30`.
Rule of three — direct proportion
Direct proportion: if quantity A is multiplied by `k`, then quantity B is also multiplied by `k`. Symbolically: `A1 / B1 = A2 / B2`.
Problem. 4 notebooks cost £6. How much do 10 notebooks cost?
Working:
- One notebook costs `6 / 4 = £1.50`.
- 10 notebooks cost `10 · 1.50 = £15`.
Rule of three — inverse proportion
Inverse proportion: if quantity A is multiplied by `k`, then quantity B is divided by `k`. The product `A · B` stays the same.
Problem. 3 workers finish a job in 12 days. How long would 6 workers take at the same pace?
Working:
- Total work = `3 · 12 = 36` worker-days.
- With 6 workers: `36 / 6 = 6` days.
How to spot direct vs. inverse?
- More of one → more of the other: direct (price, fuel, mass).
- More of one → less of the other: inverse (workers and days, speed and time).