Percentages Toolkit

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Notes

Basic Percentages

**Percentage** means 'out of $100' (e.g., 50\% = \frac{50}{100})$ .
Convert a fraction to a percentage by finding an equivalent fraction with denominator 100, or by using a calculator: $e.g., \frac{234}{650} = 0.36 = 36\%$ .
Key building blocks: **50%** (halve), **25%** (halve twice), **10%** (divide by 10), **1%** (divide by 100).
Combine blocks: $e.g., 20\% = 10\% \times$ 2; $5\% = 10\% \div$ 2; $12\% = 10\% + 1\% + 1\%$ .
With a calculator, use a **multiplier** (decimal equivalent): e.g., 12% of $650 = 0.12 \times 650 = 78$ .
To express one number as a percentage of another, write as a fraction then convert to decimal and multiply by 100.

Percentage Increases & Decreases

To **increase** by a percentage, find the percentage of the original and add it on, or find $(100 + x)\%$ and use multiplier $(1 + \frac{x}{100})$ .
To **decrease** by a percentage, find the percentage and subtract, or find $(100 - x)\%$ and use multiplier $(1 - \frac{x}{100})$ .
For repeated percentage changes, multiply all multipliers together: e.g., increase by $10\% (\times 1.10)$ then discount $15\% (\times 0.85)$ gives combined multiplier $1.10 \times 0.85 = 0.935$ .
Percentage change $= (after - before) /$ before × 100; positive = increase, negative = decrease.
Percentage $profit/loss = (selling$ price - cost price) / cost price × 100; positive = profit, negative = loss.

Reverse Percentages

A **reverse percentage** question gives the value after a percentage change and asks for the original (before) value.
Identify the multiplier **p**: for an increase of $x\%, p = 1 +$ x/100; for a decrease of $x\%, p = 1 - \frac{x}{100}$ .
Use the equation: before $\times p =$ after, so before = after / p.
Common mistake: a 10% increase followed by a 10% decrease does NOT return to the original (e.g., 100 → 110 → 99).

Percentage as a Fraction of 100

Percentage Increase and Decrease Multipliers

Reverse Percentage Flowchart

Percentage Change Formula

Practice questions

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1.What is 12% of 650?
Easy
- A78
- B65
- C72
- D80
2.Write $1.92 as a percentage of $1.60.
Easy
- A120%
- B83.3%
- C80%
- D125%
3.In a box of 80 glasses, 3 are broken. Work out the percentage of broken glasses.
Easy
- A3.75%
- B2.5%
- C4%
- D3%
4.Work out 5% of $25.
Easy
- A$1.25
- B$1.00
- C$1.50
- D$0.75
5.Work out $1.20 as a percentage of $16.
Easy
- A7.5%
- B8%
- C6%
- D9%
6.67 of 645 parcels are damaged. Calculate the percentage of parcels that are damaged.
Easy
- A10.4%
- B9.8%
- C11.2%
- D10.8%
7.Increase 42 by 16%.
Medium
- A48.72
- B46.2
- C49.2
- D47.8
8.The surface area of France is $6.41 \times 10^{5} km^{2}$ and China is $9.60 \times 10^{6} km^{2}$ . Find France's area as a percentage of China's.
Medium
- A6.68%
- B7.5%
- C5.9%
- D8.1%

Summary

## Percentages Toolkit ### Basic Percentages - **Percentage** means "out of 100". - To find a percentage of an amount without a calculator: use 50%, 25%, 10%, 1% as building blocks. - With a calculator: use a **multiplier** (decimal equivalent). E.g., 12% = 0.12; 126% = 1.26. - To express one number as a percentage of another: write as a fraction, convert to decimal, multiply by 100. ### Percentage Increases & Decreases - **Increase by x%**: new amount = original × (1 + x/100). Multiplier > 1. - **Decrease by x%**: new amount = original × (1 - x/100). Multiplier < 1. - **Percentage change**: - Multiplier method: m = after / before. (m > 1 = increase, m < 1 = decrease) - Formula: % change = (after - before) / before × 100. (positive = increase, negative = decrease) - **Percentage profit/loss**: - Multiplier: m = selling price / cost price. (m > 1 = profit, m < 1 = loss) - Formula: % profit = (selling price - cost price) / cost price × 100. - For repeated percentage changes, multiply all multipliers together first. ### Reverse Percentages - Given the final amount after a percentage change, find the original amount. - Let multiplier p = 1 ± (percentage/100). - Use: before × p = after → before = after / p. - Common mistake: a 10% increase followed by a 10% decrease does NOT return to the original. ### Key Formulas - **Multiplier for increase**: 1 + (percentage/100) - **Multiplier for decrease**: 1 - (percentage/100) - **Percentage change**: (new - original) / original × 100 - **Reverse percentage**: original = new / multiplier

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 increase or decrease an amount. For an increase of x%, multiplier = 1 + x/100; for a decrease of x%, multiplier = 1 - x/100.
Reverse percentage: Finding the original amount before a percentage change, given the final amount. Formula: original = final / multiplier.
Percentage change: The relative change between an old and new value, expressed as a percentage. Formula: (new - old) / old × 100.
Percentage profit/loss: Profit or loss expressed as a percentage of the cost price. Formula: (selling price - cost price) / cost price × 100.

Slides

What is a Percentage?

'Per-cent' means 'out of $100' (or \div 100)$ .
Percentages allow easy comparison of fractions.
Example: $\frac{1}{2} = \frac{50}{100} = 50\%, \frac{2}{5} = \frac{40}{100} = 40\%$ .
Percentages are also equivalent to decimals: $100\% = 1, 10\% = 0.1, 1\% = 0.01$ .

What is $\frac{3}{4}$ as a percentage?

Finding a Percentage of an Amount (Non-Calculator)

50%: halve the amount.
25%: halve twice (quarter).
10%: divide by 10.
1%: divide by 100.
Use these as building blocks: $e.g., 20\% = 10\% \times 2, 5\% = 10\% \div 2$ .

To find 5% of 40, first find 10% which $is ____$ , then halve it to $get ____$ .

Finding a Percentage of an Amount (Calculator)

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

What is 17% of 200?

Expressing One Number as a Percentage of Another

Write as a fraction, then convert to decimal, then multiply by 100.
Example: 7 out of 20 → $\frac{7}{20} = 0.35 = 35\%$ .
Alternatively, find an equivalent fraction with denominator 100.

Write 15 as a percentage of $60: \frac{15}{60} = ____ = ____\%$ .

Percentage Increase & Decrease (Non-Calculator)

Increase: find the percentage of the amount, then add to original.
Decrease: find the percentage, then subtract from original.
Example: Increase 30 by 10% → 10% of $30 = 3, 30+3=33$ .
Example: Decrease 30 by 10% → 10% of $30 = 3, 30-3=27$ .

Decrease 80 by 15%. What is the result?

Percentage Increase & Decrease (Calculator)

Use a multiplier: increase by 15% → multiplier $= 1.15 (100\%+15\%)$ .
Decrease by 15% → multiplier $= 0.85 (100\%-15\%)$ .
Multiply the original amount by the multiplier.

Increase 200 by 21%: multiplier = ____, result = ____.

Finding a Percentage Change

Multiplier method: $m =$ amount after / amount before.
$m > 1$ means increase, $m < 1$ means decrease.
Alternatively: percentage change $= (after - before)/before \times 100$ .
Positive = increase, negative = decrease.

A number changes from 250 to 310. What is the percentage change?

24% increase
24% decrease
19.4% increase
19.4% decrease

Reverse Percentages

Given the final amount after a percentage change, find the original.
Identify the multiplier p: increase by 4% → p=1.04; decrease by 5% → $p=0.95$ .
Use: before $\times p =$ after, so before = after / p.

After a 5% pay rise, salary is £31,500. Original salary = £31,500 / ____ = £____.

Percentage Profit and Loss

$Profit/loss = (selling$ price - cost price)/cost price $\times 100$ .
Positive = profit, negative = loss.
Multiplier method: $m =$ selling price / cost price; $m>1$ profit, $m<1$ loss.

Which of these are true?

A multiplier of 1.2 means a 20% increase.
A multiplier of 0.8 means a 20% increase.
If cost price = $50 and selling price = $60, profit $= 20\%$ .
If cost price = $100 and selling price = $80, loss $= 20\%$ .

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