Mathematics Year 7: Percentage problems: discounts, increases and more than 100%
Mathematics, Year 7 (Portugal): Turn a discount or an increase into a single percentage of the starting value, and explain your method so others can check it.
The starting value is always 100%
In a percentage problem, first decide the starting value: that is the 100%. With a 7% discount you pay 100% − 7% = 93% of the price, so the final price is 0.93 × price. With a 20% increase you end up with 100% + 20% = 120%, that is, 1.20 × the value. A percentage can go above 100%: 150% of 40 is 60, and double a quantity is 200% of it.
Explaining is part of solving
There is more than one right way to solve the same problem, and picking the most efficient one saves work: with a 7% discount you can work out 7% of the price and subtract, or go straight to 93%. So others can check it, a good explanation says four things: what the 100% is, what is asked, the calculation you did and what the result means. A bare number cannot be checked.
A 7% discount, two paths
A jacket costs €80 with a 7% discount. What do you pay? Path A: the discount is 7% of 80 = 0.07 × 80 = €5.60; you pay 80 − 5.60 = €74.40. Path B: you pay 100% − 7% = 93% of the price; 0.93 × 80 = €74.40. Answer: you pay €74.40, because the starting price (100%) drops by 7%. Both paths agree, but B needs only one calculation.
Percentages do not add up along the way
A price of €100 rises 20% and, the next month, rises 20% again. Many people think “20% + 20% = 40%” and write €140. But the second rise applies to the new price: 100 × 1.20 = 120 and 120 × 1.20 = €144, a total rise of 44%. For the same reason, a 20% drop followed by a 20% rise does not bring you back: 100 → 80 → 96. At every step, ask: 100% of what?
Sales, areas and growing amounts
In a sale, “−30%” means you pay 70% of the price. To compare areas, if a park is 3,000 m² and a garden is 2,000 m², the park is 3,000 ÷ 2,000 = 1.5 = 150% of the garden: above 100%, because it is bigger. And with an allowance that rises 20% a month, €1 becomes €1.20, €1.44, €1.73… in a spreadsheet you see at once how each month builds on the one before.
Test what you learned
A jumper costs €50 and has a 12% discount. Which calculation gives the price you pay?
- a) 88% × 50
- b) 12% × 50
- c) 112% × 50
- d) 50 − 12
Correct answer: a) 88% × 50 — Yes: you pay 100% − 12% = 88% of the price, and 0.88 × 50 = €44.
The school library has 90 adventure books this year, which is 150% of the number it had last year. How many did it have last year?
- a) 135
- b) 45
- c) 140
- d) 60
Correct answer: d) 60 — Yes: 150% = 90, so 50% = 90 ÷ 3 = 30 and 100% = 2 × 30 = 60. Check: 150% of 60 = 90.
An item costing €50 goes up 20% in May and, in June, goes up 20% again on the May price. What does it cost in June?
- a) €60
- b) €70
- c) €72
Correct answer: c) €72 — Yes: 50 × 1.20 = 60 and 60 × 1.20 = 72. Two 20% rises in a row make a 44% rise in total (72 ÷ 50 = 1.44), not 40%.
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