Olympiad topic · P5–P6
Percentages and mixtures
Common in the Hua Xia Cup's P6 first round and finals: discounts, a rise followed by a fall, and mixtures.
Try one
★★ MediumThe lunch set at a cha chaan teng costs $50.
When costs go up, the price is raised by 20%.
A month later, to win back customers, the price at that time is cut by 20%.
What does the lunch set cost now?
- Hint 1: The second change of 20% — a percentage of which price?
- Hint 2: First find the price after the rise, then treat that as the new 100%.
- 1.The rise is a percentage of the original $50: $50 × (100 + 20) ÷ 100 = $50 × 120 ÷ 100 = $60.
- 2.The cut is a percentage of $60: $60 × (100 − 20) ÷ 100 = $60 × 80 ÷ 100 = $48.
- ⚠️Common mistake: thinking a 20% rise and then a 20% cut gets back to $50. The second 20% is of $60, not $50, so the price ends $2 below where it started.
Answer: 48 dollars
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsDiscounts, strength.
- ★★MediumFree3 question types · unlimited questionsUp then down in price, mixing two cups, boiling off water.
- ★★★HardFree3 question types · unlimited questionsAdding sugar to reach a strength, cost and marked price.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsSeveral changes, swapping between cups, profit on a whole batch.
Percentages and mixtures: what it is and how to solve it
Percentages: discounts, price rises and profit, and how strong sugar or salt water is.
For example: Blocks priced at $340 are sold at 85% of the price. How much now? 340 × 85% = $289.
The key idea: First be sure what each percentage is a percentage of. Strength = sugar (or salt) ÷ the whole solution × 100%. Adding or boiling off water leaves the sugar the same; mixing two cups adds the sugar and the water separately.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download