# Percentage Problems and Simple Interest

> Write one amount as a percentage of another, work out discounts, increases and what is left, and find simple interest on savings and loans.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/percentage-problems/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The bike sale
Sam has saved 200 dollars for a bike that costs 240 dollars. Then the shop puts up a sign: "25% off everything."

"Is that enough?" Sam asks. His mum works it out: 25% is a quarter, and a quarter of 240 is 60. The bike now costs $240 - 60 = 180$ dollars. Sam has 20 dollars to spare.

"And if you put the 20 dollars in the bank," says Mum, "it will slowly earn interest. That is the bank paying you a percentage for keeping your money."

**In short:** To write an amount as a percentage, make a fraction of the whole and multiply by 100. A discount takes a percentage off and an increase adds one on. Simple interest is the same percentage of the starting amount every year: interest = amount × rate × years.

## Before you start

- Changing between fractions, decimals and percentages, and finding a percentage of an amount (see Fractions, Decimals and Percentages).
- Simplifying fractions (see Types of Fractions and Equivalent Fractions).

## One amount as a percentage of another

### Key idea: Fraction, then times 100
$$\text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\%$$

### Worked example: Test marks
Peter got 18 out of 20. $\frac{18}{20} = \frac{90}{100} = 90\%$.

In a car park of 50 vehicles, 14 are vans. The rest, 36, are cars. $\frac{36}{50} = \frac{72}{100} = 72\%$ are cars.

### Exam tip: What is left
All of something is 100%. If 78% of a tank is full, $100\% - 78\% = 22\%$ is empty.

Mr Wu spends 24% of his pay on transport and 36% on food. He has $100\% - 60\% = 40\%$ left.

## Discounts and increases

### Key idea: Two ways
- **Discount** of 25% on 240 dollars: 25% of 240 is 60, so pay $240 - 60 = 180$ dollars. Or pay 75% of 240, which is 180 dollars.
- **Increase** of 15% on 80 dollars: 15% of 80 is 12, so the new price is 92 dollars. Or find 115% of 80, which is 92.

### Watch out: Up and down is not the same
A price of 200 dollars goes up 10% to 220 dollars, then down 10%. The second 10% is of 220, which is 22, so the price ends at 198 dollars, not 200.

## Percentages of percentages

40% of a class are girls, and 25% of the girls wear glasses. The girls who wear glasses are 25% of 40%, which is $\frac{1}{4} \times 40\% = 10\%$ of the class.

## Simple interest

When you save in a bank, the bank pays you **interest**. When you borrow, you pay interest. With **simple interest**, it is the same amount every year.

### Key idea: The simple interest formula
$$I = P \times R \times T$$

- $P$ is the starting amount (the principal).
- $R$ is the rate per year, as a decimal or fraction ($5\% = 0.05$).
- $T$ is the time in years.

### Worked example: Saving 2000 dollars
2000 dollars is saved at 5% a year for 3 years.

One year's interest is 5% of 2000, which is 100 dollars. Three years: $I = 2000 \times 0.05 \times 3 = 300$ dollars.

At the end there is $2000 + 300 = 2300$ dollars.

### Challenge: Selective level: several steps
A car park has 200 vehicles. 40 are motorbikes. A quarter of the rest are vans. The rest is $200 - 40 = 160$, so there are 40 vans. That is $\frac{40}{200} = 20\%$.

After a 20% discount, a jacket costs 96 dollars. 96 dollars is 80% of the old price, so 1% is 1.20 dollars and 100% is 120 dollars.

How many years does 3000 dollars at 4% simple interest take to earn 600 dollars? One year earns 120 dollars, and $600 \div 120 = 5$ years.

**Mind map**

```text
Percentage problems
  As a percentage
    Part over whole x 100
    What is left: 100% minus
  Changes
    Discount: take off
    Increase: add on
    Up 10% then down 10% is less
  Percent of a percent
    Multiply the two
  Simple interest
    I = P x R x T
    Same each year
    Add to get the total
```

## Quick summary

- Percentage of a whole $= \frac{\text{part}}{\text{whole}} \times 100\%$.
- All of something is 100%, so what is left is 100% minus the used part.
- A 25% discount means you pay 75%. A 15% increase means you pay 115%.
- Going up 10% then down 10% does not get you back where you started.
- Simple interest: $I = P \times R \times T$, the same amount each year.
- To find the starting price after a discount, work back from the percentage you paid.

## Quiz

Try these questions from the chapter quiz.

1. Peter got 18 out of 20 questions right. What percentage is that?
   - A. 18%
   - B. 80%
   - C. 90%
   - D. 95%
   - Answer: C. 90%. $\frac{18}{20} = \frac{90}{100} = 90\%$.

2. There are 50 vehicles. 14 are vans and the rest are cars. What percentage are cars?
   - A. 14%
   - B. 28%
   - C. 36%
   - D. 72%
   - Answer: D. 72%. There are 36 cars. $\frac{36}{50} = \frac{72}{100}$.

3. A bike costs 240 dollars. It has 25% off. What is the new price?
   - A. 60 dollars
   - B. 215 dollars
   - C. 180 dollars
   - D. 200 dollars
   - Answer: C. 180 dollars. 25% of 240 is 60. $240 - 60 = 180$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/percentage-problems/#quiz