Percentage Applications

1Summary 重點摘要

Discount, interest, growth rate — percentage used in daily life.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. % to decimalConverting a percentage to a decimal is division by 100: 50% = 0.5; 25% = 0.25; 8% = 0.08; 120% = 1.2. The symbol % means 'divide by 100', so 8% is not 8 but 0.08. Backwards, a decimal becomes a percentage by multiplying by 100: 0.35 = 35%. A common error is computing with 8 instead of 8%, e.g. writing 200 × 8% as 200 × 8 = 1600, a hundred times too big; convert to a decimal or fraction before multiplying and the error cannot happen.
2. 2. Discount: 20% offWorking a discount: an item originally $200 at 20% off — paying 80% — sells for 200 × 0.8 = $160, saving $40. Note that '20% off' and 'paying 80%' describe the same event. Another: 25% off means paying 75%, and 200 × 0.75 = $150. A common error is reading 20% off as paying 20%, computing 200 × 0.2 = $40 — the price and the saving have swapped places; remember, the discount figure is the fraction paid, and the reduction is 100% minus that.
3. 3. Discount: buy 2 get 1 freeBuy 2 get 1 free: a pen costs $10; buying 2 and getting 1 free means paying for two and receiving three — 2 × 10 = $20, so the average per pen = 20 ÷ 3 ≈ $6.67, about 67% of the original price, i.e. roughly a 33% saving. Two common errors: assuming the average is 10 ÷ 2 = $5 (forgetting that $20 buys 3 pens); or writing 20 ÷ 3 as exactly $6 instead of about $6.67. The method: compute the total actually paid first, then divide by the number actually received.
4. 4. Simple interestThe simple interest formula: interest = principal × annual rate × years. Example: $1000 deposited at 5% a year for 2 years earns interest = 1000 × 0.05 × 2 = $100, and the total = 1000 + 100 = $1100. Note the rate must be a decimal — 5% becomes 0.05. Common errors: forgetting the years, giving only $50 of interest; or confusing the total with the interest. The check: $100 is exactly a tenth of $1000 — 5% a year for two years — so the figures make sense.
5. 5. Growth rateThe growth rate formula: growth rate = (new − original) ÷ original × 100%. Example: 500 students last year, 550 this year — the rise is 550 − 500 = 50, and the growth rate = 50 ÷ 500 × 100% = 10%. The denominator is the original (last year's 500), never the new value. A common error is computing 50 ÷ 550 ≈ 9.1% and calling it the growth rate; the check multiplies the original by (1 + 10%) to get 550, back to this year's number, confirming 10%. The same formula covers falls, giving a negative result or a percentage drop.
6. 6. Find original priceFinding the original price from the discounted price uses division. Example: an item sells for $160 at 20% off, i.e. original × 0.8 = 160, so original = 160 ÷ 0.8 = $200. Verify: 200 × 0.8 = 160, back to the selling price, correct. A common error is computing 160 × 0.8 = 128 — multiplying where division is needed; think about the direction: original-to-price multiplies by 0.8, so back to the original divides by 0.8. These 'find the original' questions appear constantly in exams and must be fluent.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• % to decimal = divide by 100
• Discount: multiply by 0.xx
• Find original = divide, not multiply
• Interest = principal × rate × years
• Growth = increase ÷ original