Lesson 3: Percentage in Real Life

§Text Passage

1Summary 重點摘要

Percentage is used everywhere: discounts, interest, and percentage increase/decrease. Master these three and you'll handle daily life easily.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Discount ProblemsThe core formula of discount questions is: sale price = original price × the percentage paid. For example, an original price of $200 sold at 25% off means paying 75% of the original: sale price = $200 × 75% = $150, saving $50. Why multiply by 75% and not 25%? Because '25% off' means paying three quarters of the original price; the 25% deducted is the saving, and the part paid plus the part saved make 100%. Another example: an $80 book at 10% off sells for $80 × 90% = $72, saving $8. A common mistake is to take '25% off' as a reduction to 25%, computing $200 × 25% = $50; the discount names the part you pay, so before calculating, be clear what percentage of the original price is to be paid.
2. Block 2: Interest ProblemsThe formula for interest questions is: interest = principal × annual rate × years. For example, deposit $5000 at an annual rate of 4% for 2 years: interest = $5000 × 4% × 2 = $400, and at maturity the amount withdrawn is $5000 + $400 = $5400. Why multiply by the years? Because this is simple interest: the interest earned each year is the same — $200 in the first year and $200 again in the second, $400 in all. The principal is the original sum deposited, and the rate must first become a decimal or fraction for the calculation, for example 4% = 0.04. A common mistake is to forget the years and give one year's interest of $200, or to confuse the total with the interest, answering $400 as the withdrawal; read whether the question asks for the interest or the total, which differ by exactly the principal.
3. Block 3: Percentage IncreaseThe formula for percentage increase is: increase = (new − old) ÷ old × 100%. For example, a population growing from 50000 to 60000 gives increase = (60000 − 50000) ÷ 50000 = 10000 ÷ 50000 = 0.2 = 20%. Why is the denominator the old value? Because 'increased by what percent' asks how large the added part is compared with the original; the value before the change is the base of comparison. Another example: monthly water use rising from 200 units to 230 units adds 30 units, an increase of 30 ÷ 200 = 0.15 = 15%. A common mistake is to divide by the new value, computing 30 ÷ 230 and getting about 13% instead of the correct 15%; for a percentage increase, the divisor is always the old value from before the change.
4. Block 4: Percentage DecreaseThe formula for percentage decrease is: decrease = (old − new) ÷ old × 100%. For example, a phone falling from $4000 to $3200 gives decrease = (4000 − 3200) ÷ 4000 = 800 ÷ 4000 = 0.2 = 20%, that is, 20% off. Why is the denominator again the old value? Because whether the change is up or down, its size is measured against the value before the change, keeping the standards for increase and decrease the same. Another example: complaints falling from 50 cases to 30 cases is a fall of 20 cases, a decrease of 20 ÷ 50 = 40%. A common mistake is to divide by the new value, computing 800 ÷ 3200 = 25% instead of the correct 20%; remember the words 'before the change' — the divisor is always the original amount, so circle the old value beside the expression before dividing.
5. Block 5: Find Original PriceGiven the sale price and the discount, find the original price by division: original price = sale price ÷ the percentage paid. For example, an item sold at 20% off for $160 means $160 is 80% of the original: original price = $160 ÷ 80% = $160 ÷ 0.8 = $200. Why divide? Because sale price = original price × 80%, and when the product and one factor are known, division restores the other factor. Check: $200 × 80% = $160, matching the question. Another example: at 25% off a sale price of $150 gives an original price of $150 ÷ 0.75 = $200. A common mistake is to multiply $160 by 80% once more, growing ever smaller; write the multiplication relation first, then decide whether to multiply or divide, and always finish with a multiplication check.
6. Block 6: Common TrapsThe easiest error in percentage change is the wrong denominator: the denominator is always the old value from before the change, never the new one. For example, a number rising from 80 to 100 has increased by 20, and the increase = 20 ÷ 80 = 25%, not 20 ÷ 100 = 20%. Why is this easy to slip on? Because the new value 100 is a tidy number that the careless hand reaches for. Verify the direction once more: falling from 100 to 80 is a decrease of 20 ÷ 100 = 20%; so the same gap of 20 gives different percentages for rising and falling — exactly why the base must be identified. A common mistake is to reuse one denominator for both directions; in every question, first write down which number is the old value, mark it beside the expression, and only then choose the divisor.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Discount: sale price = original × discount %
• Interest: interest = principal × rate × years
• Increase % = (new - old) ÷ old × 100%
• Decrease % = (old - new) ÷ old × 100%
• Denominator is always the OLD value