Percent word problems
In grade 7 it is no longer enough to compute one percent. Real-life situations often chain two or three percent changes together. The most important rule is: percentages do not add — the factors (100 ± p) / 100 multiply.
Type 1: discount + tax
Problem. A jacket costs £400. The shop offers a 25 % discount; at the till 20 % VAT is added. What is the final price?
Working:
- Price after discount: `400 · (100 − 25) / 100 = 400 · 0.75 = £300`.
- Price after tax: `300 · (100 + 20) / 100 = 300 · 1.20 = £360`.
Careful — the answer is not the same as treating the change as `−25 + 20 = −5 %`. That would give £380. Always multiply the factors.
Type 2: percentage increase
Problem. A club had 250 members last year. This year the number grew by 20 %. How many members does it have now?
Working:
- Increase = `250 · 20 / 100 = 50`
- New total = `250 + 50 = 300` members.
Equivalent one-liner: `250 · (100 + 20) / 100 = 250 · 1.2 = 300`.
Type 3: percentage decrease
Problem. A factory used 800 kWh of electricity last month. After an upgrade it used 25 % less. How many kWh did it save?
Working:
- Saved = `800 · 25 / 100 = 200 kWh`.
If we wanted the new usage instead: `800 · (100 − 25) / 100 = 600 kWh`.
Common traps
- Adding the percentages instead of multiplying the factors for two consecutive changes.
- Mixing up "how much less" (the saving) with "how much is left" (the new amount).
- For multistep problems, forgetting that each percent applies to the most recent value, not the original.