Percentages

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Notes

Percentages of Amounts

**Percentage** means 'out of $100' (e.g., 50\% = \frac{50}{100})$ .
To find 50%, **halve** the amount; to find 25%, halve twice; to find 10%, divide by 10; to find 1%, divide by 100.
Use these as building blocks: $e.g., 20\% = 10\% \times$ 2; $5\% = 10\% \div 2$ .
With a calculator, use a **multiplier** $(decimal equivalent): e.g., 12\% = 0.12$ , so 12% of $650 = 0.12 \times 650 = 78$ .
To express one number as a percentage of another, write as a fraction, convert to decimal, then multiply by 100.

Percentage Increases & Decreases

To increase by a percentage, find the percentage and **add** to the original. E.g., increase 30 by 10%: 10% of $30 = 3$ , $new = 33$ .
Equivalently, find $(100 + increase)\%$ of the original using multiplier $>1$ . E.g., increase 80 by 15%: multiplier 1.15, $new = 1.15 \times 80 = 92$ .
To decrease by a percentage, find the percentage and **subtract** from the original. E.g., decrease 30 by 10%: 10% of $30 = 3$ , $new = 27$ .
Equivalently, find (100 – decrease)% of the original using multiplier $<1$ . E.g., decrease 80 by 15%: multiplier 0.85, $new = 0.85 \times 80 = 68$ .

Percentage Change

Percentage change $= (After$ – Before) / Before $\times 100$ . Positive = increase, negative = decrease.
Alternatively, find multiplier $m =$ After / Before. $m > 1$ means increase; $m < 1$ means decrease.
Percentage profit/loss uses cost price and selling price: $m =$ Selling Price / Cost Price, or % profit $= (SP$ – $CP) / CP \times 100$ .
A positive % profit means profit; a negative % profit means loss.

Worked Examples (from source)

Shade 35% of a 40-square grid: $10\% = 4, 30\% = 12, 5\% = 2$ , total 14 squares shaded.
Express $150 deposit as % of $1200: $\frac{150}{1200} = 0.125 = 12.5\%$ .
Increase 200 kg by 21%: multiplier 1.21 → 242 kg.
Discount 35% on £500: multiplier 0.65 → £325.
Students from 250 to 310: change $= (310-250)/250 \times 100 = 24\%$ increase.
Car sold for £5600 from £8000: loss $= (5600-8000)/8000 \times 100 = -30\% (30\% loss)$ .

Percentage as parts of 100

Percentage increase and decrease multipliers

Practice questions

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1.Write down the fraction of the rectangle that is shaded. (The rectangle has 10 equal squares, 3 shaded.)
Easy
- A $\frac{3}{10}$
- B $\frac{1}{3}$
- C $\frac{3}{7}$
- D $\frac{10}{3}$
2.Find the percentage of the rectangle that is not shaded. (The rectangle has 10 equal squares, 3 shaded.)
Easy
- A70%
- B30%
- C3%
- D7%
3.In 2019 the rent for a café was $7275. In 2020 the rent was $7566. Calculate the percentage increase in the rent between 2019 and 2020.
Medium
- A4%
- B3.8%
- C4.2%
- D3.6%
4.There are 480 cabins on a ship. On one cruise, 456 of these cabins were used. Find the percentage of cabins that were used.
Easy
- A95%
- B90%
- C85%
- D80%
5.Last year the cost of a cruise was $4600. This year the cost of the same cruise is $4784. Work out the percentage increase in the cost.
Medium
- A4%
- B3.8%
- C4.2%
- D3.6%
6.A plane ticket costs $680 plus a tax of 16%. Find the total cost of this ticket.
Easy
- A$788.80
- B$696
- C$789
- D$108.80
7.Work out 5% of $25.
Easy
- A$1.25
- B$1.50
- C$0.50
- D$2.50
8.Work out 142 as a percentage of 304.
Medium
- A46.7%
- B46.5%
- C47%
- D46.2%

Summary

## Percentages - Quick Revision ### Key Definitions - **Percentage**: "per cent" means "out of 100". - **Decimal equivalents**: 100% = 1, 10% = 0.1, 1% = 0.01, 25% = 0.25. ### Finding a Percentage of an Amount - **Without calculator**: Use building blocks: - 50%: halve - 25%: halve twice - 10%: divide by 10 - 1%: divide by 100 - Combine: e.g., 20% = 10% × 2; 5% = 10% ÷ 2; 12% = 10% + 1% × 2. - **With calculator**: Use multiplier (decimal form of percentage). - e.g., 12% of 650 = 0.12 × 650 = 78. ### Expressing One Number as a Percentage of Another - Write as fraction, convert to decimal, then multiply by 100. - e.g., 7 as % of 20: 7/20 = 0.35 = 35%. ### Percentage Increase/Decrease - **Increase**: Add the percentage to original. Multiplier = 1 + (percentage/100). - e.g., increase 80 by 15%: 1.15 × 80 = 92. - **Decrease**: Subtract the percentage. Multiplier = 1 - (percentage/100). - e.g., decrease 80 by 15%: 0.85 × 80 = 68. ### Percentage Change - **Formula**: $ \text{Percentage Change} = \frac{\text{After} - \text{Before}}{\text{Before}} \times 100 $ - Positive = increase, negative = decrease. - **Multiplier method**: $ m = \frac{\text{Amount after}}{\text{Amount before}} $. - e.g., 1.05 → 5% increase; 0.75 → 25% decrease. ### Percentage Profit/Loss - **Formula**: $ \text{Percentage Profit} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100 $ - Positive = profit, negative = loss. - **Multiplier method**: $ m = \frac{\text{Selling Price}}{\text{Cost Price}} $. - e.g., 1.05 → 5% profit; 0.75 → 25% loss. ### Common Exam Tips - Check if answer makes sense (profit vs loss). - For discounts, find the reduced percentage (e.g., 35% off → pay 65%).

Key terms

Percentage: A number or ratio expressed as a fraction of 100, denoted by the symbol '%'.
Multiplier: The decimal equivalent of a percentage used to find a percentage of an amount or to increase/decrease by a percentage.
Percentage Increase: The amount by which a value increases, expressed as a percentage of the original value.
Percentage Decrease: The amount by which a value decreases, expressed as a percentage of the original value.
Percentage Change: The difference between the new and original values expressed as a percentage of the original value. Formula: (After - Before)/Before × 100.
Percentage Profit: Profit expressed as a percentage of the cost price. Formula: (Selling Price - Cost Price)/Cost Price × 100.
Percentage Loss: Loss expressed as a percentage of the cost price. Formula: (Cost Price - Selling Price)/Cost Price × 100 (or negative profit).

Slides

What is a Percentage?

‘Per-cent’ means ‘out of 100’ (or ‘÷ 100’).
Percentages make fractions easier to compare.
For example, $\frac{1}{2} = 50\%, \frac{2}{5} = 40\%, \frac{3}{4} = 75\%$ .

What does ‘per cent’ mean?

Out of 10
Out of 100
Out of 1000
Out of 1

Finding Percentages of Amounts (Non-Calculator)

50%: halve the amount.
25%: halve twice (find a quarter).
10%: divide by 10.
1%: divide by 100.
Use these as building blocks: $e.g., 20\% = 10\% \times 2$ .

To find 5% of an amount, find 10% and then $____ it$ .

Finding Percentages of Amounts (Calculator)

Use a multiplier: the decimal equivalent of the percentage.
$E.g., 12\% = 0.12$ , so 12% of $650 = 0.12 \times 650 = 78$ .
For percentages $> 100\%$ , multiplier $> 1 (e.g., 126\%$ → 1.26).

What is 8% of 140?

11.2
8.4
10.8
12.6

Expressing One Number as a Percentage of Another

Write the first number as a fraction of the second.
Find the decimal equivalent (using calculator if needed).
Multiply by 100 to get the percentage.
E.g., 7 out of $20 = \frac{7}{20} = 0.35 = 35\%$ .

Work out $1.20 as a percentage of $16. Answer: ____%

Percentage Increase (Non-Calculator)

Find the percentage of the original amount.
Add it to the original amount.
E.g., Increase 30 by 10%: 10% of $30 = 3, 30 + 3 = 33$ .
This is equivalent to finding 110% of the amount.

Increase 42 by 16%. What is the result?

48.72
58.8
46.2
52.08

Percentage Increase (Calculator)

Use a multiplier: $100\% + increase\% = new$ percentage.
Convert to decimal: e.g., increase by 15% → $115\% = 1.15$ .
Multiply original by multiplier: $1.15 \times 80 = 92$ .

The price of a television is $560. It is reduced by 18% in a sale. The sale price is $____.

Percentage Decrease (Calculator)

Use a multiplier: $100\% - decrease\% = new$ percentage.
Convert to decimal: e.g., decrease by 15% → $85\% = 0.85$ .
Multiply original by multiplier: $0.85 \times 80 = 68$ .

A plane ticket costs $680 plus a tax of 16%. What is the total cost?

$788.80
$789.60
$790.00
$785.60

Percentage Change

Multiplier method: $m =$ amount after / amount before.
Formula: % change $= (after - before) /$ before $\times 100$ .
Positive answer = increase, negative = decrease.

The number of passengers on a train increases from 63 to 77. The percentage increase $is ____\%$ .

Percentage Profit and Loss

$Profit/loss =$ selling price - cost price.
Multiplier: $m =$ selling price / cost price.
Formula: % profit $= (selling - cost) /$ cost $\times 100$ .
Positive = profit, negative = loss.

Roberto buys a toy for $5.00 and sells it for $4.60. What is his percentage loss?

Which of these are true?

To increase by 20%, multiply by 1.2.
To decrease by 20%, multiply by 0.8.
A multiplier of 1.05 means a 5% decrease.
A multiplier of 0.95 means a 5% decrease.

Flashcards

What does 'per cent' mean?

Out of $100 (or \div 100)$ .

How do you convert a decimal to a percentage?

Multiply by 100.

How do you convert a percentage to a decimal?

Divide by 100.

How do you find 50% of an amount without a calculator?

Halve the amount.

How do you find 25% of an amount without a calculator?

Halve the amount twice (find a quarter).

How do you find 10% of an amount without a calculator?

Divide the amount by 10.

How do you find 1% of an amount without a calculator?

Divide the amount by 100.

How do you find 5% of an amount without a calculator?

Find 10% and halve it.

How do you find 20% of an amount without a calculator?

Find 10% and double it.

How do you find 12% of an amount without a calculator?

Find 10% and 1%, then add the 10% and two lots of the 1%.

How do you find a percentage of an amount with a calculator?

Write the percentage as a decimal multiplier and multiply by the amount.

What is the multiplier for 12%?

0.12

What is the multiplier for 126%?

1.26

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

Write the first number as a fraction of the second, convert to a decimal, then multiply by 100.

How do you increase an amount by a percentage without a calculator?

Find the percentage of the amount and add it to the original amount.

How do you increase an amount by a percentage with a calculator?

Find the multiplier for the new percentage $(100\% + increase\%)$ and multiply by the original amount.

What multiplier do you use to increase by 15%?

1.15

How do you decrease an amount by a percentage without a calculator?

Find the percentage of the amount and subtract it from the original amount.

How do you decrease an amount by a percentage with a calculator?

Find the multiplier for the remaining percentage $(100\% - decrease\%)$ and multiply by the original amount.

What multiplier do you use to decrease by 15%?

0.85

How do you calculate percentage change using the multiplier method?

$m =$ amount after / amount before. Then $m = 1 + (percentage change/100)$ for increase, or $m = 1 - (percentage change/100)$ for decrease.

How do you calculate percentage change using the formula?

Percentage Change $= (After - Before) /$ Before $\times 100$ . Positive result is an increase, negative is a decrease.

How do you calculate percentage profit or loss using the multiplier method?

$m =$ selling price / cost price. $m > 1$ indicates profit, $m < 1$ indicates loss.

How do you calculate percentage profit or loss using the formula?

Percentage Profit $= (Selling$ Price - Cost Price) / Cost Price $\times 100$ . Positive is profit, negative is loss.

Work out 5% of $25.

$1.25

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