Percentages of amounts and percentage problems

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Lesson notes

What is a Percentage?

A percentage is a part out of 100. The symbol is %.
50% means 50 out of 100, or half.
You can write a percentage as a fraction with denominator 100. For example, 25% = 25/100.
You can also write a percentage as a decimal. For example, 25% = 0.25.
To change a percentage to a decimal, divide by 100. To change a decimal to a percentage, multiply by 100.

Finding a Percentage of an Amount

To find 10% of a number, divide the number by 10. For example, 10% of 80 is 8.
To find 1% of a number, divide the number by 100. For example, 1% of 500 is 5.
To find 25% of a number, divide the number by 4. For example, 25% of 60 is 15.
To find 50% of a number, divide the number by 2. For example, 50% of 120 is 60.
For other percentages, you can use a calculator: multiply the number by the percentage as a decimal. For example, 15% of 200 is 200 × 0.15 = 30.

Using Mental Math for Common Percentages

You can find 10% first, then use it to get other percentages.
To find 20%, find 10% and double it. For example, 20% of 50: 10% is 5, so 20% is 10.
To find 5%, find 10% and halve it. For example, 5% of 80: 10% is 8, so 5% is 4.
To find 15%, find 10% and add 5%. For example, 15% of 60: 10% is 6, 5% is 3, so 15% is 9.
To find 75%, find 25% and multiply by 3. For example, 75% of 40: 25% is 10, so 75% is 30.

Finding the Whole When You Know a Percentage

If you know that 20% of a number is 30, you can find the whole.
First, find 1% by dividing the given amount by the percentage. For example, 30 ÷ 20 = 1.5, so 1% is 1.5.
Then multiply by 100 to get the whole. So 1.5 × 100 = 150.
Another way: write an equation. Let x be the whole. Then 20% of x = 30, so 0.20 × x = 30. Solve: x = 30 ÷ 0.20 = 150.
Check: 20% of 150 is 30, so it works.

Converting Between Fractions, Decimals, and Percentages

To change a fraction to a percentage, divide the top by the bottom, then multiply by 100. For example, 3/4 = 0.75 = 75%.
To change a percentage to a fraction, write it over 100 and simplify. For example, 40% = 40/100 = 2/5.
To change a decimal to a percentage, multiply by 100. For example, 0.6 = 60%.
To change a percentage to a decimal, divide by 100. For example, 8% = 0.08.
Percentages can be more than 100% or less than 1%. For example, 150% = 1.5, and 0.5% = 0.005.

Percentage Increase and Decrease

Percentage increase means the amount goes up. For example, if a price goes from $50 to $60, the increase is $10.
To find the percentage increase, divide the increase by the original amount, then multiply by 100. So $10 ÷ $50 = 0.2, then 0.2 × 100 = 20%.
Percentage decrease means the amount goes down. For example, if a price goes from $80 to $60, the decrease is $20.
To find the percentage decrease, divide the decrease by the original amount, then multiply by 100. So $20 ÷ $80 = 0.25, then 0.25 × 100 = 25%.
You can also find the new amount after a percentage change. For a 10% increase on $200, find 10% of $200 = $20, then add: $200 + $20 = $220.

Real-Life Uses: Discounts, Sales, Tips, and Interest

A discount is a percentage taken off the original price. For example, a 20% discount on a $30 toy means you save $6, so you pay $24.
A sale price is the original price minus the discount. To find it, calculate the discount amount and subtract.
A tip is extra money you give for good service, often a percentage of the bill. For example, a 15% tip on a $40 meal is $6, so you pay $46.
Interest is money you earn or pay, often a percentage of an amount. For example, 5% interest on $100 in a bank gives you $5 after a year.
Always read the problem carefully to see if you need to add or subtract the percentage amount.

Comparing Percentages of Different Totals

Percentages let you compare parts of different-sized wholes. For example, 50% of 10 is 5, and 50% of 100 is 50.
A bigger percentage does not always mean a bigger amount if the totals are different. For example, 20% of 200 is 40, but 50% of 50 is 25.
To compare, you can find the actual amounts and then compare them.
Sometimes you need to find what percentage one number is of another. For example, 15 is what percent of 60? Divide 15 by 60 to get 0.25, then multiply by 100 to get 25%.
This helps in real life, like comparing test scores from different classes.

Percentage Change Formula

Percentage change tells you how much something has gone up or down as a percentage of the original.
The formula is: percentage change = (change ÷ original) × 100%.
Here, change means new value minus original value. If the change is positive, it's an increase; if negative, it's a decrease.
For example, if a value goes from 40 to 50, the change is 10, so percentage change = (10 ÷ 40) × 100% = 25% increase.
This formula works for any positive original value. If the original is zero, you cannot find a percentage change.

Tips for Solving Percentage Problems

Always identify the whole (the total amount) and the part (the amount you are comparing).
Use the equation: part = percentage (as a decimal) × whole.
If you know the part and the percentage, you can find the whole by dividing: whole = part ÷ percentage (as a decimal).
If you know the part and the whole, you can find the percentage by dividing and multiplying by 100: percentage = (part ÷ whole) × 100%.
Check your answer by estimating. For example, 30% of 50 should be about 15, since 30% is close to 1/3.

Slides

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Practice questions

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1.What does the percent sign (%) mean?
Easy
- Aout of 100
- Bout of 10
- Cout of 50
- Dout of 1000
2.What is 50% of 80?
Easy
- A40
- B30
- C50
- D20
3.A jacket was originally $80. Its price increased by 15%. What is the new price?
Medium
- A$92
- B$95
- C$88
- D$90
4.Which of the following are equal to 25%? (select all that apply)
Easy
- A0.25
- B1/4
- C2/5
- D25/100
- E0.025
5.50% of 60 is 30.
Easy
True or false?
6.Match each fraction to its equivalent percentage.
Medium
- 1/2
- 1/4
- 3/4
- 50%
- 25%
- 75%
7.Put these percentages in order from smallest to largest: 15%, 5%, 25%, 10%.
Medium
- 15%
- 5%
- 25%
- 10%
8.Put these percentages in order from smallest to largest: 10%, 25%, 5%, 50%.
Easy
- 10%
- 25%
- 5%
- 50%

Revision notes

## Percentages of Amounts and Percentage Problems ### Finding a Percentage of a Quantity - To find a percentage of a number, turn the percentage into a fraction or decimal and multiply. - **10%** of a number = divide by 10. For example, 10% of 80 = 8. - **1%** of a number = divide by 100. For example, 1% of 500 = 5. - **25%** of a number = divide by 4. For example, 25% of 60 = 15. - **50%** of a number = divide by 2. For example, 50% of 40 = 20. - With a calculator, you can do: 15% of 200 = 0.15 × 200 = 30. ### Finding the Whole When a Percentage Is Known - If you know a part and its percentage, you can find the whole. - Example: If 20% of a number is 40, then 1% is 40 ÷ 20 = 2, so 100% is 2 × 100 = 200. - Or use the formula: whole = part ÷ (percentage as a decimal). So, 40 ÷ 0.20 = 200. ### Converting Between Fractions, Decimals, and Percentages - **Percent** means 'out of 100'. So, 45% = 45/100 = 0.45. - To convert a fraction to a percentage, divide top by bottom and multiply by 100. For example, 3/4 = 0.75 = 75%. - To convert a decimal to a percentage, multiply by 100. For example, 0.6 = 60%. - Percentages can be more than 100% (e.g., 150% = 1.5) or less than 1% (e.g., 0.5% = 0.005). ### Percentage Increase and Decrease - **Increase**: new value = original + (percentage of original). - **Decrease**: new value = original − (percentage of original). - Example: A shirt costs $40 and is on sale for 25% off. The discount is 25% of $40 = $10, so the sale price is $40 − $10 = $30. - Example: A bank pays 5% interest on $200. The interest is 5% of $200 = $10, so the new amount is $200 + $10 = $210. - **Tip**: To find the new amount directly, multiply by (1 + percentage as a decimal) for increase, or (1 − percentage as a decimal) for decrease. For 25% off, multiply by 0.75. ### Comparing Quantities as Percentages of Different Totals - To compare, find what percentage each part is of its own total, then compare the percentages. - Example: In class A, 15 out of 30 students like maths. That's 50%. In class B, 20 out of 50 like maths. That's 40%. So class A has a higher percentage. ### Key Points - Percentages are a way to compare parts to a whole of 100. - Always think: 'What is the whole? What is the part?' - Practice with real-life examples like shopping discounts, test scores, and interest.

percentage: A number out of 100, shown with the % sign. For example, 25% means 25 out of 100.
percent: Another word for percentage. It means 'out of 100'.
fraction: A part of a whole, written with a top number and a bottom number, like 1/2.
decimal: A number with a decimal point, like 0.5. It can also show parts of a whole.
increase: To become larger. A percentage increase makes a number bigger.
decrease: To become smaller. A percentage decrease makes a number smaller.
discount: A reduction in price, often given as a percentage off.
interest: Money paid for using someone else's money, often a percentage of the amount saved or borrowed.
tip: Extra money given to a worker, like a waiter, often a percentage of the bill.
whole: The total amount. When finding a percentage, the whole is the number you start with.
part: A piece of the whole. When finding a percentage, the part is the amount you get.

Flashcards

What does 'percent' mean?

'Percent' means 'out of 100'. So 25% means 25 out of 100.

How do you write 0.45 as a percentage?

Multiply by 100: 0.45 × 100 = 45%. So 0.45 = 45%.

How do you write 3/4 as a percentage?

First convert to a decimal: 3 ÷ 4 = 0.75. Then multiply by 100: 0.75 × 100 = 75%. So 3/4 = 75%.

What is 10% of 80?

10% means 10 out of 100, which is 0.10. Multiply: 0.10 × 80 = 8. So 10% of 80 is 8.

What is 1% of 500?

1% means 1 out of 100, which is 0.01. Multiply: 0.01 × 500 = 5. So 1% of 500 is 5.

What is 25% of 60?

25% is the same as 1/4. Divide 60 by 4: 60 ÷ 4 = 15. So 25% of 60 is 15.

What is 50% of 90?

50% is the same as 1/2. Divide 90 by 2: 90 ÷ 2 = 45. So 50% of 90 is 45.

How do you find 20% of a number without a calculator?

First find 10% by dividing by 10, then double it. For example, 20% of 40: 10% is 4, double is 8. So 20% of 40 is 8.

What is 15% of 200?

Find 10% (20) and 5% (10), then add: 20 + 10 = 30. So 15% of 200 is 30.

If 30% of a number is 15, what is the number?

Divide 15 by 30% (0.30): 15 ÷ 0.30 = 50. So the number is 50.

A shirt costs $40 and is on sale for 25% off. What is the sale price?

Find 25% of $40: 0.25 × 40 = $10. Subtract from the original: $40 - $10 = $30. The sale price is $30.

How do you convert a percentage to a decimal?

Divide by 100. For example, 35% becomes 0.35. Or move the decimal point two places to the left.

What is 150% of 80?

150% is 1.50 as a decimal. Multiply: 1.50 × 80 = 120. So 150% of 80 is 120.

What is 0.5% of 200?

0.5% is 0.005 as a decimal. Multiply: 0.005 × 200 = 1. So 0.5% of 200 is 1.

A price increases from $50 to $60. What is the percentage increase?

The change is $60 - $50 = $10. Divide by the original: $10 ÷ $50 = 0.20. Multiply by 100: 20%. So it's a 20% increase.

A price decreases from $80 to $60. What is the percentage decrease?

The change is $80 - $60 = $20. Divide by the original: $20 ÷ $80 = 0.25. Multiply by 100: 25%. So it's a 25% decrease.

What is the formula for percentage change?

Percentage change = (change ÷ original) × 100. Change is new value minus original value.

If a bank gives 5% interest on $200, how much interest do you earn?

5% of $200 is 0.05 × 200 = $10. You earn $10 interest.

If you leave a 15% tip on a $30 meal, how much is the tip?

15% of $30 is 0.15 × 30 = $4.50. The tip is $4.50.

Which is larger: 20% of 50 or 30% of 40?

20% of 50 = 10, and 30% of 40 = 12. So 30% of 40 is larger.

What is the relative change from 10 to 15?

Change is 15 - 10 = 5. Divide by original: 5 ÷ 10 = 0.5. Multiply by 100: 50%. So it's a 50% increase.

What does 'percentage error' mean?

Percentage error compares a measured value to the true value. It is (difference ÷ true value) × 100.

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