Percentages

플레이하며 배우기

이 문제들을 풀어 에너지를 얻은 뒤 낚시하고 탐험하세요. 계정이 필요 없어요.

교육자를 위해: Percentages(KS3 Maths, Number)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트 — 수업에 사용하거나, 학습자들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.

수업 노트

What a Percentage Means

A percentage is a number or ratio written as a fraction of 100.
The word comes from the Latin per centum, meaning 'by the hundred'.
The symbol is %, and 45% means 45 out of every 100, which is the fraction 45/100 or the decimal 0.45.
A percentage is a pure (dimensionless) number, so it has no units.
Percentages express a proportionate part of a total, so always ask what the total (100%) is.
Values can be above 100% or below 0%: 111% and −35% are both valid, especially for changes and comparisons.

Converting Between Percentages, Fractions and Decimals

To turn a percentage into a decimal, divide by 100: 25% = 25/100 = 0.25.
To turn a decimal into a percentage, multiply by 100: 0.06 = 6%.
To turn a percentage into a fraction, write it over 100 and simplify: 45% = 45/100 = 9/20.
Do not divide by 100 and keep the % sign at the same time: 25% = 0.25, but 25%/100 would wrongly mean 0.0025.
A term like 100/100% is incorrect because it would be read as 1%.

Finding a Percentage of an Amount

To find a percentage of an amount, convert the percentage to a decimal and multiply: 50% of 500 = 0.50 × 500 = 250.
Mentally, first find 10% by dividing by 10, then 1% by dividing by 100, and build from these.
For example, 15% = 10% + 5%, and 5% is half of 10%.
Because multiplication is commutative, 50% of 20 equals 20% of 50; both give 10.
To find a percentage of a percentage, multiply the decimals: 50% of 40% = 0.50 × 0.40 = 0.20 = 20%.

Expressing One Quantity as a Percentage of Another

Write the first quantity as a fraction of the second, then multiply by 100.
Example: 50 apples out of 1,250 apples is 50/1250 = 0.04, which is 4%.
You can multiply by 100 first instead: 50 × 100 = 5,000, then 5,000 ÷ 1,250 = 4%.
Always state what the total (100%) is, because a percentage only makes sense relative to that total.

Percentage Increase and Decrease

A '10% rise' or '10% fall' is usually relative to the initial value.
If an item costs $200 and rises by 10%, the increase is $20 and the new price is $220.
The new price is 110% of the initial price, because 100% + 10% = 110%.
A 100% increase doubles the quantity: the final amount is 200% of the initial amount.
An 800% increase makes the final amount 9 times the original, since 100% + 800% = 900%.

Using Multipliers

A multiplier is the decimal you multiply by to apply a percentage change in one step.
For a 10% increase, the multiplier is 1.10, because the result is 110% of the original.
For a 10% decrease, the multiplier is 0.90, because the result is 90% of the original.
In general, an increase of p% uses the multiplier 1 + p/100, and a decrease of p% uses 1 − p/100.
Example: a 15% increase on 80 is 80 × 1.15 = 92.

Percentage Change

Percentage change compares the change to the original amount.
An increase of $0.15 on a price of $2.50 is the fraction 0.15/2.50 = 0.06, which is a 6% increase.
The method is: divide the change by the original amount, then multiply by 100.
The original amount is the 100% reference point for the change.

Reverse Percentages

A reverse percentage question gives the result after a change and asks for the original amount.
Use the multiplier backwards: if the final amount is 110% of the original, divide by 1.10.
Mentally, first work out what 1% corresponds to, then scale up to 100%.
Example: if 42 kg is 7%, then 1% is 42 ÷ 7 = 6 kg, so 100% is 600 kg.
In reverse-percentage work, W is the percentage part, p% is the rate, and G is the basic (original) value.

Simple Interest and Real-Life Contexts

Simple interest is calculated on the original amount only, not on interest already earned.
Percentages are used for discounts, VAT, pay rises, profit and loss, and interest rates.
Historically, interest rates were quoted in hundredths by the 17th century, which is why we use percentages today.
When solving a real-life percentage problem, identify the original amount, the rate, and whether it is an increase or decrease.

슬라이드

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

가입 — 무료 로그인

연습 문제

무료 미리 보기 — 65개 중 8개 문제. 가입하면 전부 볼 수 있어요.

1.Which decimal is equal to 45%?
Easy
- A0.45
- B4.5
- C0.045
- D45
2.50 apples are taken from a total of 1250 apples. What percentage of the total is this?
Medium
- A4%
- B0.04%
- C25%
- D40%
3.What is 50% of 40%?
Medium
- A20%
- B10%
- C90%
- D200%
4.In a college, 60% of students are female and 5% of female students are computer science majors. What percentage of all students are female computer science majors?
Medium
- A3%
- B5%
- C30%
- D65%
5.In the same college, 10% of all students are computer science majors. What percentage of computer science majors are female?
Medium
- A30%
- B3%
- C50%
- D60%
6.50% of 20 is equal to 20% of 50.
Easy
True or false?
7.What is 15% of 80?
Easy
- A12
- B15
- C8
- D120
8.A jacket costs £60. In a sale its price is reduced by 20%. What is the sale price?
Medium
- A£48
- B£40
- C£52
- D£12

더 많은 문제

9.Match each percentage to its equivalent decimal.
Easy
- 45%
- 7%
- 120%
- 0.5%
- 0.45
- 0.07
- 1.2
- 0.005
10.Arrange these values in ascending order, from smallest to largest.
Medium
- 0.08
- 18%
- 0.8
- 80%
11.A percentage is a number or ratio expressed as a fraction of 100.
Easy
True or false?
12.Select all statements that are correct. (select all that apply)
Medium
- A45% = 0.45
- B25% = 0.025
- C50% of 40% = 20%
- DA 100% increase doubles a quantity
- EPercentages must always be between 0 and 100
13.What is 35% of 240?
Medium
- A84
- B8.4
- C74
- D96
14.A jumper is reduced by 20% in a sale and now costs £36. What was the original price?
Medium
- A£43.20
- B£45
- C£56
- D£40
15.A price rises from $200 to $220. Select all true statements. (select all that apply)
Medium
- AThe increase is $20
- BThe percentage increase is 10%
- CThe new price is 110% of the original price
- DThe percentage increase is 20%
16.A student scores 18 out of 24 in a test. What is this as a percentage?
Medium
- A75%
- B72%
- C80%
- D133%
17.What is 4% of 250?
Medium
- A10
- B100
- C1
- D62.5
18.A phone costs £240. Its price is reduced by 12.5%. What is the new price?
Medium
- A£210
- B£30
- C£227.50
- D£252
19.A population of 800 increases by 5%. What is the new population?
Medium
- A840
- B805
- C760
- D4000
20.A jacket is reduced by 20% in a sale and now costs £64. What was the original price?
Medium
- A£80
- B£76.80
- C£84
- D£128
21.Which of the following are equal to 0.6? (select all that apply)
Medium
- A3/5
- B60%
- C6%
- D0.60
- E6/100
22.To find 3% of a number, you can multiply the number by 0.03.
Easy
True or false?
23.A rectangle has length 20 cm and width 15 cm. Its length is increased by 10% and its width is decreased by 10%. What is the area of the new rectangle?
Medium
- A297 cm²
- B300 cm²
- C303 cm²
- D330 cm²
24.A shop buys a coat for £80 and sells it for £100. What is the percentage profit?
Medium
- A20%
- B25%
- C12.5%
- D80%
25.A car is bought for £12,000. Its value depreciates by 15% in the first year and by 10% in the second year. What is its value after two years?
Hard
- A£9,180
- B£9,000
- C£10,200
- D£9,180,000
26.A shirt is priced at £30. In a sale, the price is reduced by 20%. At the till, a further 10% discount is taken off the sale price. What is the final price?
Medium
- A£21.00
- B£21.60
- C£24.00
- D£20.00
27.A circle has radius 5 cm. Its radius is increased by 20%. By what percentage does the area increase?
Medium
- A20%
- B40%
- C44%
- D144%
28.A triangle has base 8 cm and height 5 cm. Its base is increased by 25%. What is the new area?
Medium
- A20 cm²
- B25 cm²
- C30 cm²
- D40 cm²
29.A rectangular garden measures 12 m by 8 m. Its length is increased by 10% and its width is decreased by 10%. Select all true statements. (select all that apply)
Medium
- AThe new length is 13.2 m.
- BThe new width is 7.2 m.
- CThe new area is 95.04 m².
- DThe area is unchanged.
- EThe new perimeter is 40.8 m.
30.A house is bought for £200,000. Its value increases by 10% in the first year and then decreases by 10% in the second year. Select all true statements. (select all that apply)
Hard
- AAfter the first year, the value is £220,000.
- BAfter the second year, the value is £198,000.
- CAfter the second year, the value is £200,000.
- DThe overall change is a decrease of 1%.
- EThe overall change is no change.
31.If a quantity is increased by 20% and then the result is decreased by 20%, the final value is the same as the original.
Easy
True or false?
32.What does the word 'percent' literally mean?
Easy
- Aper hundred
- Bper thousand
- Cper ten
- Dper part
33.Which symbol is used to denote a percentage?
Easy
- A%
- B‰
- Cpct
- Dpc
34.What is 45% as a decimal?
Easy
- A0.45
- B4.5
- C0.045
- D45
35.Which of the following is the correct way to calculate 25% of 80?
Medium
- A80 × 0.25
- B80 ÷ 0.25
- C80 × 25
- D80 ÷ 25
36.A price increases from $200 to $220. What is the percentage increase?
Medium
- A10%
- B20%
- C5%
- D12%
37.Which of the following statements about percentages are true? (select all that apply)
Medium
- AA percentage is a dimensionless number.
- BPercentages can be negative.
- CPercentages can be greater than 100.
- DPercentages are always between 0 and 100.
- EThe percent sign evolved from the Italian term 'per cento'.
38.Percentages can be greater than 100.
Easy
True or false?
39.The abbreviation 'pct' is sometimes used instead of the percent sign.
Easy
True or false?
40.In a college, 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female?
Hard
- A30%
- B3%
- C50%
- D20%
41.An item's price rises by $0.15 from $2.50. What is the percentage increase? Give your answer as a number without the % sign.
Easy
42.After a 15% increase, a train ticket costs £92. What was the original price? Give your answer in pounds.
Medium
43.Complete the sentence about the meaning of percent.
Easy
The word percent comes from the Latin per centum, meaning 'by the ____'.
44.A coat costs £80. In a sale its price is reduced by 15%. What is the sale price in pounds?
Medium
45.A bicycle is priced at $250. Its price then rises by 8%. What is the new price in dollars?
Medium
46.An amount increases by 10%, then the new amount is decreased by 10%. Fill in the blanks.
Medium
The final amount is ____% of the original amount, so overall the amount ____.
47.£500 is invested at a simple interest rate of 4% per year. How much interest is earned in 3 years, in pounds?
Medium
48.A shirt costs £45. Its price is reduced by 10%. What is the new price in pounds?
Medium
49.Complete the sentence: To find 25% of a number, divide the number by ___.
Medium
To find 25% of a number, divide the number by ____.
50.A class has 500 students and 50% of them are male. How many students are male?
Medium
51.In a college, 60% of all students are female, and 5% of female students are computer science majors. What percentage of all students are female computer science majors? Give your answer as a number without the % sign.
Medium
52.In the same college, 10% of all students are computer science majors. What percentage of computer science majors are female? Give your answer as a number without the % sign.
Medium
53.Explain why writing 25%/100 is incorrect, and state the correct decimal value of 25%.
Medium
54.In a school, 60% of students are female and 10% of all students study French. If 5% of female students study French, what percentage of French students are female? Explain your reasoning.
Hard
55.Explain how to find 30% of 80 without a calculator.
Medium
56.A rectangle has length 8 cm and width 5 cm. The length is increased by 25% and the width is decreased by 20%. Explain whether the area changes, showing your working.
Hard
57.An item's price rises by $0.15 from $2.50. What is the percentage increase? Give your answer as a number without the % sign.
Easy
- A7 %
- B6 %
- C5 %
- D8 %
58.After a 15% increase, a train ticket costs £92. What was the original price? Give your answer in pounds.
Medium
- A96 £
- B80 £
- C72 £
- D88 £
59.A coat costs £80. In a sale its price is reduced by 15%. What is the sale price in pounds?
Medium
- A68 £
- B82 £
- C75 £
- D61 £
60.A bicycle is priced at $250. Its price then rises by 8%. What is the new price in dollars?
Medium
- A243 $
- B270 $
- C324 $
- D297 $
61.£500 is invested at a simple interest rate of 4% per year. How much interest is earned in 3 years, in pounds?
Medium
- A72 £
- B54 £
- C66 £
- D60 £
62.A shirt costs £45. Its price is reduced by 10%. What is the new price in pounds?
Medium
- A40.5 pounds
- B44.5 pounds
- C48.5 pounds
- D36.5 pounds
63.A class has 500 students and 50% of them are male. How many students are male?
Medium
- A250 students
- B225 students
- C300 students
- D275 students
64.In a college, 60% of all students are female, and 5% of female students are computer science majors. What percentage of all students are female computer science majors? Give your answer as a number without the % sign.
Medium
- A2 %
- B5 %
- C4 %
- D3 %
65.In the same college, 10% of all students are computer science majors. What percentage of computer science majors are female? Give your answer as a number without the % sign.
Medium
- A33 %
- B27 %
- C36 %
- D30 %

복습 노트

## Percentages **Percent** means "by a hundred" (Latin *per centum*). A percentage is a number or ratio expressed as a fraction of 100, written with the **%** sign. - 45% = $\frac{45}{100}$ = 0.45 - 50% of 500 students = 250 students - Percentages are dimensionless (pure numbers) but are used to express proportions. - Values can go outside 0–100: 111% or −35% are possible. ## Converting - Fraction → percentage: multiply by 100. - Decimal → percentage: multiply by 100. - Percentage → decimal: divide by 100 (e.g. 25% = 0.25). **Never** divide by 100 and keep the % sign: 25% = 0.25, not 25%/100. ## Finding a percentage of an amount Convert the percentage to a decimal or fraction, then multiply. - 20% of 60 = 0.20 × 60 = 12 - Mental method: find 10% (divide by 10), then 5% (half of 10%), then 1% (divide by 100). ## Expressing one quantity as a percentage of another Divide the part by the whole, then multiply by 100. - 50 apples out of 1250: $\frac{50}{1250} = 0.04$, so 4%. ## Percentage of a percentage Convert both to decimals and multiply. - 50% of 40% = 0.50 × 0.40 = 0.20 = 20%. ## Increase and decrease (multipliers) A change is usually relative to the **initial** value. - Increase by 10%: multiply by 1.10 (100% + 10% = 110%). - Decrease by 10%: multiply by 0.90. - Increase of 100% doubles the amount (200% of original). - Increase of 800% makes it 9 times as large (100% + 800% = 900%). Example: $200 rises 10% → $200 × 1.10 = $220. ## Percentage change \[ \text{percentage change} = \frac{\text{change}}{\text{original}} \times 100 \] Example: $0.15 increase on $2.50 → $\frac{0.15}{2.50} = 0.06$ = 6% increase. ## Reverse percentages To find the original amount, divide by the multiplier. - 42 kg is 7% of what? $\frac{42}{0.07} = 600$ kg. - If a price after a 20% rise is $96, original = $\frac{96}{1.20} = \$80$. ## Simple interest \[ I = P \times r \times t \] (I = interest, P = principal, r = rate as a decimal, t = time in years) ## Real-life contexts - **Discounts**: reduce price by a percentage. - **VAT**: a tax added as a percentage. - **Pay rises**: percentage increase in salary. Always state what the percentage is **relative to** (the total that is 100%).

Percentage: A number or ratio expressed as a fraction of 100, denoted by the % sign.
Percent: From Latin per centum, meaning 'by a hundred'.
Multiplier: The decimal factor used to increase or decrease an amount, e.g. 1.10 for a 10% increase, 0.90 for a 10% decrease.
Percentage change: The change in a quantity expressed as a percentage of the original value: (change ÷ original) × 100.
Reverse percentage: Finding the original amount when the final amount and the percentage change are known, by dividing by the multiplier.
Simple interest: Interest calculated as I = P × r × t, where P is principal, r is the rate as a decimal, and t is time in years.
VAT: Value Added Tax, a percentage tax added to prices.
Discount: A reduction in price by a given percentage.

플래시카드

What does the word 'percent' literally mean?

It comes from the Latin per centum, meaning 'by the hundred'. So a percentage is a number or ratio expressed as a fraction of 100.

Write 45% as a fraction and as a decimal.

45% = 45/100 = 0.45.

How do you find 50 apples as a percentage of 1,250 apples?

Divide the part by the whole, then multiply by 100: 50/1250 = 0.04, and 0.04 × 100 = 4%.

What is 50% of 40%?

Convert both to decimals and multiply: 0.50 × 0.40 = 0.20 = 20%.

Is it correct to write 25%/100? Explain.

No. 25% already means 25/100, so 25%/100 would mean 25/100/100 = 0.0025, which is wrong. Write 25% = 25/100 = 0.25.

Why is it important to state what a percentage is relative to?

A percentage is always a proportion of some total that corresponds to 100%. Without knowing that total, the percentage has no clear meaning.

In a college, 60% of students are female and 5% of females study computer science. What percentage of all students are female computer science majors?

60/100 × 5/100 = 3/100, so 3% of all students are female computer science majors.

In that same college, 10% of all students are computer science majors. What percentage of computer science majors are female?

3% of all students are female computer science majors, so 3%/10% = 30/100 = 30% of computer science majors are female.

Is 50% of 20 the same as 20% of 50? Why?

Yes, both equal 10. Multiplication is commutative, so reversing the two percentages does not change the result.

How do you find a percentage of an amount using a multiplier?

Convert the percentage to a decimal and multiply by the amount. For example, 35% of 80 = 0.35 × 80 = 28.

How do you express one quantity as a percentage of another?

Divide the first quantity by the second, then multiply by 100. For example, 15 out of 60 = 15/60 × 100 = 25%.

What multiplier do you use for a 10% increase?

1.10, because the final amount is 100% + 10% = 110% of the original, and 110% = 1.10.

What multiplier do you use for a 15% decrease?

0.85, because the final amount is 100% − 15% = 85% of the original, and 85% = 0.85.

An item costs $200 and the price rises by 10%. What is the new price, and what percentage of the original is it?

The increase is 10% of $200 = $20, so the new price is $220. This is 110% of the original price.

What does an increase of 100% do to a quantity?

The final amount is 200% of the initial amount, so the quantity has doubled.

What does an increase of 800% do to a quantity?

The final amount is 100% + 800% = 900% of the original, which is 9 times as large.

How do you calculate percentage change?

Percentage change = (change ÷ original amount) × 100. For example, a rise of $0.15 on $2.50 is 0.15/2.50 = 0.06 = 6%.

How do you find the original amount after a percentage change (reverse percentage)?

Write the final amount as a multiplier of the original, then divide. For example, if a price after a 20% increase is $96, then 1.20 × original = 96, so original = 96 ÷ 1.20 = $80.

42 kg is 7% of a total. How do you find the total (100%)?

First find 1%: 42 ÷ 7 = 6 kg. Then multiply by 100: 6 × 100 = 600 kg.

What is simple interest, and how is it calculated?

Simple interest is interest calculated only on the original amount. It equals original × rate × time, where the rate is a decimal.

A sum of $500 earns simple interest at 4% per year for 3 years. How much interest is earned?

500 × 0.04 × 3 = $60.

How do you find the price of an item after a discount?

Subtract the discount from 100% and use that as a multiplier. For example, a 30% discount means paying 70%, so multiply the original price by 0.70.

How do you add VAT at 20% to a price?

Multiply the price by 1.20, since the total is 120% of the original price.

Can a percentage be greater than 100 or negative? Give an example.

Yes. Percentages are not restricted to 0–100; values like 111% or −35% are common, especially for percent changes and comparisons.

What is the difference between a percentage and a fraction?

A percentage is a specific fraction with denominator 100. For example, 45% = 45/100, while a fraction such as 3/5 can be written as 60%.

Unlock all 65 questions, flashcards & more

무료 계정을 만들어 이 주제의 모든 문제, 슬라이드, 플래시카드, 복습 노트를 확인하세요.

가입 — 무료 로그인

기출 문제

이 주제의 기출 문제 연습이 곧 나와요.

곧 출시