Lesson 5: Advanced Percentage

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1Summary 重點摘要

Top-level percentage: tax, profit and loss, compound interest. These are common P6 exam types. Master them and you've finished the percentage unit.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Tax CalculationTotal with tax = price × (1 + tax rate). For example, a price of $500 with a tax rate of 8% gives total = $500 × 1.08 = $540, of which the tax is $40. Why multiply by 1.08 rather than 0.08? Because the total has two parts: the original 100% of the price plus the 8% of tax, together 108%, which is the multiplier 1.08. Given the total, recover the price by division: price = $540 ÷ 1.08 = $500. Another example: goods of $200 under a 10% tax come to $200 × 1.1 = $220 with $20 of tax. A common mistake is to answer with the tax alone, giving $40 as the total; read whether the question asks for the tax or the total with tax — they differ by exactly the price — and circle the key word before answering.
2. Block 2: Profit PercentageThe formula for profit percentage is: profit % = (selling price − cost) ÷ cost × 100%. For example, cost $200 sold for $250 gives profit = (250 − 200) ÷ 200 = 50 ÷ 200 = 0.25 = 25%, a gain of 25%. Why is the denominator the cost and not the selling price? Because gains and losses are measured against the money put in: the same profit of $50 means entirely different rates on a cost of $200 (25%) and on a cost of $500 (only 10%). Another example: goods costing $400 sold for $460 give profit = 60 ÷ 400 = 15%. A common mistake is to divide by the selling price, computing 50 ÷ 250 = 20% instead of the correct 25%; in a profit percentage the divisor is always the cost, so circle the cost before writing the expression.
3. Block 3: Loss PercentageThe formula for loss percentage is: loss % = (cost − selling price) ÷ cost × 100%. For example, cost $200 sold for $160 gives loss = (200 − 160) ÷ 200 = 40 ÷ 200 = 0.2 = 20%, a loss of 20%. Why is the denominator still the cost? Because the loss measures the damage relative to the money invested, and the cost is the base at the start of the deal — the same base as the profit percentage, which keeps the two comparable. Another example: goods costing $500 sold for $450 give a loss of 50 ÷ 500 = 10%. A selling price above the cost is profit, below it is loss; the two formulas mirror each other, and the numerator is in both cases the difference between the selling price and the cost. A common mistake is to reverse the numerator and get a negative number; first judge profit or loss, then pick the matching formula.
4. Block 4: Simple InterestThe formula for simple interest is: interest = principal × rate × years. For example, a principal of $1000 at 4% a year for 3 years earns interest = $1000 × 4% × 3 = $120, and the total is $1000 + $120 = $1120. Why 'simple'? Because every year uses the original principal $1000, the interest never compounds, each year brings $40, and three years make three shares found by one multiplication. Compare the 'interest per year' line of a bank book: with simple interest the yearly amount is identical and can be checked at a glance. A common mistake is to forget the years and give a single year's $40, or to mix up the interest with the total, answering $120 where $1120 belongs; first see whether the question asks for the interest or the total, and then write the answer.
5. Block 5: Compound InterestThe formula for compound interest is: total = principal × (1 + rate) to the power of the years. For example, a principal of $1000 at 4% a year grows in 3 years to $1000 × (1.04)³ = $1000 × 1.124864 = $1124.86. Why does it differ from simple interest? Because compound interest adds each year's interest to the principal and computes the next year on the new amount: the first year brings $40 and the principal becomes $1040; the second year brings $41.60 and the principal grows again. Interest on interest rolls ever larger, so after 3 years the total exceeds the simple-interest $1120 by $4.86. A common mistake is to compute compound interest as 4% × 3 = 12%; compound interest must be worked year by year or by a power, and 'rate × years' belongs to simple interest alone — the two scopes must not be mixed.
6. Block 6: Common TrapsThe most frequent trap in profit and loss is the wrong denominator: the denominator is always the cost, never the selling price. For example, cost $80 sold for $100 gives profit = 20 ÷ 80 = 25%, not 20 ÷ 100 = 20%. The other trap is mixing compound with simple interest: at 5% a year for 3 years, simple interest is 5% × 3 = 15%, while compound interest is (1.05)³ − 1 = 15.76% — not equal, because compound interest adds interest to interest every year. Why do tests love these two traps? Because they probe the understanding of bases and compounding, and only a clear concept avoids them. A common mistake is to multiply rate by years on sight; before writing anything, judge whether the interest is simple or compound and whether the base is the cost or the selling price, and then choose the formula.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Total with tax = price × (1 + tax rate)
• Profit % = (sell - cost) ÷ cost × 100%
• Loss % = (cost - sell) ÷ cost × 100%
• Simple interest = principal × rate × years
• Compound interest = principal × (1 + rate)^years