Percentages Review

1Summary 重點摘要

Review percentages learned in primary school, know how to convert percentages to decimals and fractions.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What is a percentageA percentage expresses how many parts out of every hundred, written with the sign %. So 50% means 50 out of 100 parts — exactly one half; 25% is one quarter; 100% is the whole, equal to 1. The power of percentages is a common standard: if 50 of 200 students in one school wear glasses and 100 of 500 in another, comparing headcounts means little, but as percentages, 25% against 20% shows at a glance which school has the higher rate. Discounts (20% off means paying 80%), battery levels and phone storage are all quoted in percentages, making it one of the most used tools in daily life.
2. Step 2: Percentage to decimalTo convert a percentage to a decimal, divide the number before the % sign by 100 — that is, move the decimal point two places left — then drop the sign. For example 25% = 25 ÷ 100 = 0.25; 150% = 150 ÷ 100 = 1.5; 6% = 6 ÷ 100 = 0.06 (note it is 0.06, not 0.6 — the point must move the full two places, filling with zeros if needed). This step underpins all percentage calculations, because multiplying requires the decimal form first: to find 6% of 200, compute 200 × 0.06 = 12, not 200 × 6. The classic mistake is moving one place only, writing 6% as 0.6 and inflating the answer tenfold.
3. Step 3: Decimal to percentageTurning a decimal into a percentage is the reverse: multiply by 100 (move the point two places right) and add the % sign. For example 0.3 = 0.3 × 100% = 30%; 1.2 = 1.2 × 100% = 120% (above 100% is perfectly sensible — it means more than one whole); 0.07 = 7%. As a check, the percentage figure must be 100 times the decimal: if 0.3 comes out as 3%, you multiplied by 10 and have gone wrong. Fractions convert via decimals first: 38 = 0.375 = 37.5%, handy for win rates and pass rates. Drill both directions until fluent.
4. Step 4: Percentage to fractionTo write a percentage as a fraction, put the number before the % sign over 100, then simplify. For example 25% = 25100 = 14; 150% = 150100 = 32, that is 1 12; 40% = 40100 = 25. Memorising a few pairs speeds up answers: 10% = 110, 20% = 15, 25% = 14, 50% = 12, 75% = 34. The benefit: for '25% of a number' you can think 'divide by 4' directly — 25% of 360 is 360 ÷ 4 = 90, faster than converting to a decimal first. Knowing both routes lets you pick whichever suits the numbers in the question.
5. Step 5: One number as a percentage of anotherTo find what percentage one quantity is of another, divide the part by the whole and multiply by 100%. For example, if 5 of 20 apples are rotten, the rotten share is 5 ÷ 20 × 100% = 25%. Another case: of the $60 in Ming's money box, $24 came from red packets, so red-packet money is 24 ÷ 60 × 100% = 40%. The key is telling part from whole: the quantity before 'of' is usually the part, and the whole goes in the denominator. The classic error is inverting the division, writing 20 ÷ 5 = 400%. Sanity-check afterwards — a part cannot exceed the whole, so a percentage over 100% (unless allowed) usually means the division was flipped.
6. Step 6: Finding a percentage of a number (forward and reverse)Distinguish the two most basic percentage applications. Forward: to find 20% of 30, multiply: 30 × 20% = 30 × 0.2 = 6. Reverse: given that 40% of a number is 20, divide: 20 ÷ 40% = 20 ÷ 0.4 = 50. The rule for choosing: given the whole and asked for the part, multiply; given the part and asked for the whole, divide. For instance, 'after a 15% discount an item sells for $170; what was the original price?' Here 85% of the original is 170, so the original is 170 ÷ 0.85 = $200. Reverse questions tempt students to multiply; let which quantity is known decide the direction.
7. Step 7: Percentages in daily lifePercentages are everywhere in daily life, and reading them well helps with money decisions. Bank interest uses an annual rate: on a $5000 deposit at 2% per year, one year's interest is 5000 × 0.02 = $100. Shop discounts: trainers marked $400 at 25% off sell for 400 × 75% = $300, saving $100. Pass rates: if 34 of 40 students pass, the pass rate is 34 ÷ 40 × 100% = 85%. A phone battery at 20% means one fifth remains. Whenever you meet such figures, first ask 'percentage of what' — whether the base is $400 or the discounted price changes the answer a great deal.
8. Step 8: Common mistakes summaryThree traps dominate percentage work. First, moving the point the wrong number of places: writing 6% as 0.6 (it is 0.06) or 0.5 as 5% (it is 50%); the fix is 'divide by 100, move two places'. Second, treating a 20% rise followed by another 20% rise as a 40% rise: starting at $100, after the first rise you hold $120, and the second 20% acts on $120, giving $144 — a real rise of 44%, because the base changed. Third, multiplying in reverse questions: if 80% of a number is 40, the number is 40 ÷ 0.8 = 50, not 40 × 0.8 = 32. Substitute your answer back into the question and confirm the logic before writing the final result.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Percentage = how many parts out of 100
• To decimal: divide by 100 (left two places)
• To percentage: multiply by 100% (right two places)
• 100% = 1
• Find percentage: Part ÷ Whole × 100%