Percentages Revision Notes for OCR GCSE Maths

Percentages

A percentage is a fraction whose denominator is 100. This means that percentages are simply another way of expressing fractions or decimals.

Percentage formula:

$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$

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Example: If you have 32 out of 100 marks, your percentage score is: $\frac{32}{100} \times 100 = 32\%$

Converting Between Percentages, Fractions, and Decimals

From Fraction to Percentage: Multiply the fraction by 100.

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Example: Convert $\frac{3}{4}$ to a percentage.

\frac{3}{4} \times 100 = 75\%

From Percentage to Fraction: Divide the percentage by 100 and simplify.

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Example: Convert 75% to a fraction.

\frac{75}{100} = \frac{3}{4}

From Percentage to Decimal: Divide the percentage by 100.

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Example: Convert 75% to a decimal.

75\%=0.75

From Decimal to Percentage: Multiply the decimal by 100.

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Example: Convert 0.75 to a percentage.

0.75×100=75\%

Percentage of an Amount

When you are asked to find a percentage of a given number, use the following steps:

Step 1: Convert the percentage to a decimal or a fraction.
Step 2: Multiply this by the amount.

Worked Example: Calculating Percentages of an Amount

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Example Problem: You have £320. What is:

(a) 15%
(b) 63%
(c) 17.5% of £320? To solve these problems, follow these steps:

Step 1: Break Down the Percentages

Start by finding percentages that are easy to calculate. Then, use these to build up the percentage you need.

10%:

\frac{10}{100} \times 320 = \frac{320}{10} = \pounds 32

\frac{1}{100} \times 320 = \frac{320}{100} = \pounds 3.20

50%:

\frac{50}{100} \times 320 = \frac{320}{2} = \pounds 160

20%:

\frac{20}{100} \times 320 = 2 \times 10\% = 2 \times 32 = \pounds 64

\frac{5}{100} \times 320 = \frac{10\%}{2} = \frac{32}{2} = \pounds 16

2.5%:

\frac{2.5}{100} \times 320 = \frac{5\%}{2} = \frac{16}{2} = \pounds 8

Step 2: Calculate the Specific Percentages

Use the values calculated above to find:

(a) 15% of £320:

15% is the sum of 10% and 5%.

15\%=10\%+5\%

32+16=£48

(b) 63% of £320:

63% can be broken down as 50%, 10%, and 3%.

63\%=50\%+10\%+3\%

First, find 3% by multiplying 1% by 3:

3\%=3×1\%=3×3.2=£9.60

160+32+9.60=£201.60

(c) 17.5% of £320:

17.5% can be broken down as 10%, 5%, and 2.5%.

17.5\%=10\%+5\%+2.5\%

32+16+8=£56

On a Calculator

When dealing with more complex percentages or when you're not using mental maths, it's important to understand how to calculate percentages using a calculator. This method is not only simple but also very reliable for non-calculator exams.

Key Concept: Percentages as Decimals

Percentages are essentially decimals in disguise. To convert a percentage into a decimal, divide by 100. Once you have the decimal, you can multiply it by the amount you're interested in to find the percentage of that amount.

Steps to Find a Percentage Using a Calculator

Convert the Percentage to a Decimal
Multiply the Amount by the Decimal

Convert the Percentage to a Decimal:

Divide the percentage by 100.

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Example: 23% becomes $\frac{100}{23}=0.23$ .

Multiply the Amount by the Decimal:

Use your calculator to perform this multiplication.

Worked Examples:

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Example 1: Find 23% of 135g

Convert to a Decimal:

23\%=\frac{23}{100}=0.23

Multiply by the Amount:

135×0.23=31.05g

Final Answer: 31.5g

Note: Remember to keep track of your units (grammes in this case).

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Example 2: Find 4% of £22.45

Convert to a Decimal:

4\%=\frac{4}{100}=0.04

Multiply by the Amount:

22.45×0.04=0.898 rounded to 2 decimal places =£0.90

Final Answer: £0.90 Note: It's important to round money to two decimal places.

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Example 3: Find 31.8% of 1,435,988

Convert to a Decimal:

31.8\%=\frac{31.8}{100}=0.318

Multiply by the Amount:

1,435,988×0.318=456,644.584 rounded to nearest whole number =456,644

Final Answer: 456,644 Note: Here, rounding to the nearest whole number is appropriate due to the large size of the figures.

Percentage Change

Percentage Change is used to determine how much an amount has increased or decreased in relation to its original value. This is particularly useful in comparing the difference between two values to see how significant the change is.

Formula:

$\text{Percentage Change} = \left(\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}}\right) \times 100$

This formula can give you either a positive percentage (indicating an increase) or a negative percentage (indicating a decrease).

Worked Examples:

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Example 1: Calculating a Percentage Increase Problem: After using a maths revision website, your mark in a maths test went from 34 to 46. What is the percentage increase?

Identify the values:

New Value = 46
Old Value = 34

Substitute into the formula:

\text{Percentage Change} = \left(\frac{\text{46} - \text{34}}{\text{34}}\right) \times 100

Calculate the difference:

46−34=12

Divide by the Old Value:

\frac{12}{34}≈0.3529

Multiply by 100:

0.3529×100≈35.3%

Final Answer: The percentage increase is 35.3%.

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Example 2: Calculating a Percentage Decrease Problem: A scientific calculator's price has been reduced from £4.99 to £3.50. What is the percentage decrease?

Identify the values:

New Value = £3.50
Old Value = £4.99

Substitute into the formula:

\text{Percentage Change} = \left(\frac{\text{3.50} - \text{4.99}}{\text{4.99}}\right) \times 100

Calculate the difference:

3.50−4.99=−1.49

Divide by the Old Value:

\frac{−1.49}{4.99}≈−0.2986

Multiply by 100:

−0.2986×100≈−29.9%

Final Answer: The percentage decrease is 29.9%.

Note: The negative sign indicates a decrease.

4. Percentage Increase

When you want to increase a number by a certain percentage, you aren't just adding that percentage as a separate value; you're scaling the entire amount. This means you multiply the original amount by a factor that represents both the original 100% and the additional percentage.

Method:

Convert the Percentage to a Decimal:

Divide the percentage by 100.

Add 1 to the Decimal:

This represents the original amount plus the increase.

Multiply the Original Amount by this New Value. Formula:

$\text{New Amount} = \text{Original Amount} \times (1 + \text{Percentage Increase as a Decimal})$

Worked Examples:

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Example 1: Increase £235 by 17%

Convert 17% to a Decimal:

17\%=\frac{17}{100}=0.17

Add 1 to the Decimal:

1+0.17=1.17

Multiply the Original Amount:

235×1.17=274.95

Final Answer: After a 17% increase, £235 becomes £274.95.

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Example 2: Increase 87kg by 3.5%

Convert 3.5% to a Decimal:

3.5\%=\frac{3.5}{100}=0.035

Add 1 to the Decimal:

1+0.035=1.035

Multiply the Original Amount:

87×1.035=90.045

Final Answer: After a 3.5% increase, 87kg becomes 90.045kg.

Percentage Decrease

When you want to decrease a number by a certain percentage, you subtract that percentage from the original amount. This means you multiply the original amount by a factor that represents the remaining percentage after the decrease.

Method:

Convert the Percentage to a Decimal:

Divide the percentage by 100.

Subtract the Decimal from 1:

This represents the remaining percentage after the decrease.

Multiply the Original Amount by this New Value. Formula:

$\text{New Amount} = \text{Original Amount} \times (1 - \text{Percentage Decrease as a Decimal})$

Worked Examples:

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Example 1: Decrease 250g by 24%

Convert 24% to a Decimal:

24\%=\frac{24}{100}=0.24

Subtract the Decimal from 1:

1−0.24=0.76

Multiply the Original Amount:

250×0.76=190

Final Answer: After a 24% decrease, 250g becomes 190g.

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Example 2: Decrease £10.20 by 64.5% 26. Convert 64.5% to a Decimal:

64.5\%=\frac{64.5}{100}=0.645

Subtract the Decimal from 1:

1−0.645=0.355

Multiply the Original Amount:

10.20 \times 0.355 = 3.621 \text{ rounded to 2 decimal places } = 3.62

Final Answer: After a 64.5% decrease, £10.20 becomes £3.62.

Compound Interest

Compound interest differs from simple interest because the amount you earn each year is based on the growing balance, not just the original principal. This means that your money can grow faster over time as the interest compounds.

Formula for Compound Interest:

$\text{Total Amount} = P \times \left(1 + \frac{r}{100}\right)^n$

Where:

$P$ is the principal amount (initial amount of money).
$r$ is the annual interest rate (as a percentage).
$n$ is the number of years the money is invested or borrowed for.

Worked Example:

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Example: The bank pays me a compound interest rate of 5% on my balance each year. At the start, I have £30 in the account. How much will I have after 25 years?

Identify the Values:

Principal ( $P$ ) = £30
Interest rate ( $r$ ) = 5%
Number of years ( $n$ ) = 25

Convert the Interest Rate:

5\% = \frac{5}{100} = 0.05

Apply the Compound Interest Formula:

\text{Total Amount} = 30 \times \left(1 + 0.05\right)^{25}

\text{Total Amount} = 30 \times (1.05)^{25}

Calculate the Result:

First, calculate $1.05^{25}$ :

1.05^{25} \approx 3.386

Then multiply by the principal:

30×3.386≈101.59

Final Answer: After 25 years, the total amount will be approximately £101.59.

Reverse Percentages

When you know the final value after a percentage change and need to find the original value, you're dealing with a reverse percentage problem. The key is to work backwards from the final value to determine what the original value was before the percentage change.

Formula:

$\text{Original Value} = \frac{\text{Final Value}}{(1 \pm \text{Percentage Change as a Decimal})}$

Use $+$ for percentage increases.
Use $-$ for percentage decreases.

Worked Example:

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Example: Your car has decreased in value by 23%. It is now worth £654.50. What was it worth before?

Identify the Information:

Final Value = £654.50
Percentage Decrease = 23%

Convert the Percentage to a Decimal:

23\%=\frac{23}{100}=0.23

Subtract the Decimal from 1 (since it's a decrease):

1−0.23=0.77

Set Up the Equation:

We know the final value after the decrease is given by:

Final Value=Original Value×0.77

So, to find the original value (w):

w×0.77=654.50

Solve for the Original Value:

w = \frac{654.50}{0.77} \approx 850

Final Answer: The car was originally worth £850.

Percentages (OCR GCSE Maths): Revision Notes

Percentages

Percentage formula:

Converting Between Percentages, Fractions, and Decimals

Percentage of an Amount

Worked Example: Calculating Percentages of an Amount

Step 1: Break Down the Percentages

Step 2: Calculate the Specific Percentages

On a Calculator

Key Concept: Percentages as Decimals

Steps to Find a Percentage Using a Calculator

Worked Examples:

Example 1: Find 23% of 135g

Example 2: Find 4% of £22.45

Example 3: Find 31.8% of 1,435,988

Percentage Change

Worked Examples:

4. Percentage Increase

Worked Examples:

Example 1: Increase £235 by 17%

Example 2: Increase 87kg by 3.5%

Percentage Decrease

Worked Examples:

Example 1: Decrease 250g by 24%

Compound Interest

Worked Example:

Example: The bank pays me a compound interest rate of 5% on my balance each year. At the start, I have £30 in the account. How much will I have after 25 years?

Reverse Percentages

Worked Example:

Example: Your car has decreased in value by 23%. It is now worth £654.50. What was it worth before?

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