Percentages

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Lesson notes

Percentages of Amounts

Percentage means 'out of 100' (e.g., 50% = 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% × 2; 5% = 10% ÷ 2.
With a calculator, use a multiplier (decimal equivalent): e.g., 12% = 0.12, so 12% of 650 = 0.12 × 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 × 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 × 80 = 68.

Percentage Change

Percentage change = (After – Before) / Before × 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 × 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: 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 × 100 = 24% increase.
Car sold for £5600 from £8000: loss = (5600-8000)/8000 × 100 = -30% (30% loss).

35% of a 40-square grid

Slides

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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
- A3/10
- B1/3
- C3/7
- D10/3
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.
Medium
- 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%

Revision notes

## 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%).

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).

Flashcards

What does 'per cent' mean?

Out of 100 (or ÷ 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 × 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 × 100. Positive is profit, negative is loss.

Work out 5% of $25.

$1.25

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