Percentages

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課程筆記

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.

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練習題

免費預覽——65 題中的 8 題。註冊即可查看全部。

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

複習筆記

## 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.

字卡

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%.

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