Percentages and Percentage Change for the ESAT

Percentage as parts per hundred

Percentage means 'number of parts per hundred'. This allows us to compare different scales by bringing them to a common base of 100. For instance, if 12 sweets out of every 50 in a mixture are red, this is equivalent to 24 sweets per hundred. Therefore, we say 24 per cent of the sweets are red, written as 24%.

Consider another example: if 3 out of every 8 pupils in a school eat more than their 5 a day of fruit and vegetables, what percentage is that? To solve this, we can scale the ratio to 100. Since 3 out of 8 is the same as 300 out of 800, we find how many that is out of 100 by dividing 300 by 8. Because $300 \div 8 = 37.5$, 3 out of 8 is 37.5 out of 100, or 37.5%.

Percentages as fractions or decimals used multiplicatively

Because 24% means 24 per hundred, it can be written as the fraction $\frac{24}{100}$ (which simplifies to $\frac{6}{25}$) or as the decimal 0.24.

When we need to calculate a percentage of a quantity, we interpret the percentage multiplicatively. Calculating 24% of 50 means multiplying 50 by 24%. To do this, we convert the percentage into a fraction or a decimal:

$\frac{24}{100} \times 50 = 12$ or $0.24 \times 50 = 12$.

Worked Example: Calculating 35% of 220

First, convert the percentage to a fraction: $35\% = \frac{35}{100} = \frac{7}{20}$

Then, multiply by the quantity: $\frac{7}{20} \times 220 = 7 \times 11 = 77$

Expressing one quantity as a percentage of another

To find what percentage one number is of another, first write the relationship as a fraction, then convert that fraction into a percentage by finding that fraction of 100.

Example 1: 18 as a percentage of 30

1. Write as a fraction: $\frac{18}{30} = \frac{3}{5}$
2. Multiply by 100: $\frac{3}{5} \times 100 = 60$
3. Result: 18 is 60% of 30.

Example 2: What percentage of 80 is 24?

1. Write as a fraction: $\frac{24}{80} = \frac{3}{10}$
2. Multiply by 100: $\frac{3}{10} \times 100 = 30$
3. Result: 24 is 30% of 80.

Comparing two quantities using percentages

Percentages allow us to compare proportions across different total amounts.

Example 1: Sweets in bags Bag A contains 200 sweets, with 54 red ones. Bag B contains 150 sweets, with 42 red ones. 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.

Example 2: Oxygen concentration Tank A has 300 ml of oxygen in 2.5 l of gas. Tank B has 840 ml of oxygen in 7.5 l of gas. To compare them, we ensure the units are the same: 2.5 l = 2500 ml and 7.5 l = 7500 ml.

Percentage in Tank A: $\frac{300}{2500} \times 100\% = 12\%$ Percentage in Tank B: $\frac{840}{7500} \times 100\% = \frac{840}{75}\% = \frac{56}{5}\% = 11.2\%$ The concentration of oxygen is greater in Tank A.

Percentages greater than 100% and percentage change

A percentage can exceed 100% when a value increases or is compared to a smaller reference. If a salary increases by 10%, the new salary is 110% of the original.

We use specific formulas to calculate changes:

$\text{Percentage change} = \frac{\text{actual change}}{\text{original amount}} \times 100$ $\text{Percentage increase or decrease} = \frac{\text{actual increase or decrease}}{\text{original amount}} \times 100$ $\text{Percentage profit or loss} = \frac{\text{actual profit or loss}}{\text{original price}} \times 100$

Worked Example: The Carpenter's Chair A carpenter makes a chair for a total cost of £22 and sells it for £77.

1. What percentage is the sale price of her cost price? $\frac{77}{22} \times 100\% = \frac{7}{2} \times 100\% = 350\%$

2. What is her percentage profit? Actual profit: $£77 - £22 = £55$ Percentage profit: $\frac{55}{22} \times 100\% = 250\%$

Finding the original price after an increase or decrease

In these problems, always treat the original price as 100%.

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

Worked Example: Price of Shoes After a 20% increase, a pair of shoes costs £96. What was the original price? If the original price was 100%, the new price is $100\% + 20\% = 120\%$. If 120% is £96, then 1% is $\frac{£96}{120}$. So 100% (the original price) is $\frac{£96}{120} \times 100 = £80$.

Simple interest

Simple interest is calculated solely on the original principal amount. If the interest is $r\%$ per annum (per year), the total simple interest gained over a period is simply the yearly interest multiplied by the number of years.

Worked Example: John's Savings John puts £400 into a savings account with 3.5% simple interest per annum. How much interest will he receive after 8 years?

Method 1: Yearly interest first Each year he receives $\frac{3.5}{100} \times £400 = £14$ in interest. In 8 years, he receives $8 \times £14 = £112$.

Method 2: Total percentage first Total interest percentage is $8 \times 3.5\% = 28\%$. 28% of £400 is $\frac{28}{100} \times £400 = £112$.