Percentages and Percentage Change for the TMUA

Percentage as parts per hundred

The term percentage literally means number of parts per hundred. For example, if a mixture contains 12 red sweets out of every 50, this is equivalent to 24 sweets per hundred. We write this as 24 per cent, or $24\%$.

In mathematical problems, percentages are interpreted multiplicatively as either fractions or decimals. $24\%$ is equivalent to $\frac{24}{100}$ (which simplifies to $\frac{6}{25}$) or $0.24$. To calculate $24\%$ of 50, you multiply the quantity by the fraction or decimal: $\frac{24}{100} \times 50 = 12$ or $0.24 \times 50 = 12$.

Example: Calculating Percentages

If 3 out of every 8 pupils in a school eat more than five portions of fruit a day, what percentage is that?

3 out of 8 can be written as $\frac{3}{8}$. To convert this to a percentage, we consider it as a fraction of 800: 300 out of 800. Dividing both by 8 gives $37.5$ out of 100. Thus, 3 out of 8 is $37.5\%$.

Example: Multiplicative Interpretation

What is $35\%$ of 220?

$35\% = \frac{35}{100} = \frac{7}{20}$.

$35\%$ of 220 is $\frac{7}{20} \times 220 = 7 \times 11 = 77$.

Expressing and Comparing Quantities

To express one quantity as a percentage of another, first write the first quantity as a fraction of the second. Ensure both are in the same units. Then, multiply this fraction by 100.

Example: Expressing One Quantity as a Percentage

To express 18 as a percentage of 30, we write $\frac{18}{30}$. This simplifies to $\frac{3}{5}$. Multiplying by 100 gives $\frac{3}{5} \times 100 = 60\%$. Therefore, 18 is $60\%$ of 30.

Example: Comparing Two Quantities

Percentages allow for a like for like comparison between groups of different sizes. Bag A contains 200 sweets with 54 red ones. Bag B contains 150 sweets with 42 red ones. Which has the higher concentration of red sweets?

For Bag A: $\frac{54}{200} = \frac{27}{100} = 27\%$.

For Bag B: $\frac{42}{150} = \frac{84}{300} = \frac{28}{100} = 28\%$.

Bag B has a higher percentage of red sweets.

Percentages Greater than 100%

Percentages can exceed 100 when a value increases beyond its original state. If a salary increases by $10\%$, the new salary is $110\%$ of the original.

Example: Large Percentages

A carpenter makes a chair for a total cost of £22 and sells it for £77. What is this as a percentage of her cost price?

Expressing 77 as a percentage of 22: $\frac{77}{22} \times 100\% = \frac{7}{2} \times 100\% = 350\%$.

Percentage Change and Profit

Percentage change measures the size of a change relative to the original amount. The formula is:

$\text{Percentage change} = \frac{\text{actual change}}{\text{original amount}} \times 100$

This applies to percentage increase, decrease, profit, or loss. In the case of profit, the original amount is the cost price.

Example: Percentage Profit

Using the carpenter example: cost is £22, sale price is £77. The actual profit is $£77 - £22 = £55$.

$\text{Percentage profit} = \frac{55}{22} \times 100\% = 250\%$.

Original Value Problems

When a value has already undergone a percentage change, you must treat the original value as $100\%$.

1. If a price decreases by $x\%$, the new price $Q$ represents $(100 - x)\%$ of the original.
2. If a price increases by $y\%$, the new price $R$ represents $(100 + y)\%$ of the original.

Example: Finding Original Price

After a percentage increase of $20\%$, the price of shoes is £96. Find the original price.

The original price is $100\%$. The new price is $100\% + 20\% = 120\%$.

If $120\% = £96$, then $1\% = \frac{96}{120}$.

Therefore, $100\% = \frac{96}{120} \times 100 = £80$.

Simple Interest Calculations

Simple interest is a fixed percentage of the original principal amount added for every time period (usually per annum). Unlike compound interest, the interest itself does not earn further interest.

Example: Simple Interest

John puts £400 into an account with $3.5\%$ simple interest per annum. How much interest will he receive after 8 years?

Method 1: Annual Calculation

Interest per year: $\frac{3.5}{100} \times £400 = £14$.

Total interest over 8 years: $8 \times £14 = £112$.

Method 2: Combined Rate

Total percentage interest over 8 years: $8 \times 3.5\% = 28\%$.

$28\%$ of £400: $\frac{28}{100} \times £400 = £112$.