Reverse percentages

Sometimes a question will ask you to work backwards and find the original price of something after the price has increased. If you are given a quantity after a percentage increase or decrease, and you need to find the original amount, use this method:

Example 1

A radio sells for £63, after a 40% increase in the cost price. Find the cost price.

Solution

Start with the original amount as 100%.

Cost price = 100%

We are told the selling price is a 40% in the cost price.

So the selling price is 100% + 40% = 140% of the cost price.

We know that the selling price is £63, so 140% = £63.

Now calculate 1%:

140% = £63

1% = £63/140

1% = £0.45

The cost price is 100%, so multiply £0.45 by 100.

Cost price = 0.45 × 100 = £45.

Example 2

A new car falls in value by 30% in a year. After a year, it is worth £8,400.

Find the price of the car when it was new.

Solution

Remember that the original price of the car is 100%.

Original price = 100%.

Second-hand price = 100% - 30% = 70%.

So £8,400 = 70% of the original price.

So 1% of original price = £8,400 ÷ 70

Original price = 100% = 100 x 1% = 100 x (£8,400 ÷ 70)

= £12,000.

It is easy to go wrong in this type of question. Always check that your answer is realistic.

Cumulative increase and decrease

Simple interest

With simple interest the amount of money borrowed remains fixed.

For example £400 is borrowed for 3 years at an interest rate of 5% pa (pa means per annum, or each year).

Interest for one year = 5% of £400

= (⁵/₁₀₀ ) × 400

= £20

Interest for 3 years = £20 × 3 = £60.

You can write this in a formula.

Interest = P × R × T

• P (principal) is the amount borrowed.
• R is the rate of interest per year.
• T is the time in years.

Compound Interest

Here the interest is added to the principal at the end of each year. So the next year the interest is worked out on a larger amount of money than what was originally borrowed.

This means paying interest on the interest of previous years (unlike simple interest, where you only pay interest on the original amount).

This is how it is calculated:

£400 is borrowed for 3 years at 5% compound interest.

Principal at the start = £400

Interest in the 1st year = ⁵/₁₀₀ × 400 = £20

Principal after 1 year = £420

Interest in the 2nd year = ⁵/₁₀₀ × 420 = £21

Principal after 2 years = £441

Interest in the 3rd year = ⁵/₁₀₀ × 441 = £22.05

Principal after 3 years = £463.05

The total interest charged under compound interest will be £63.05.

This is different to the simple interest worked out above.

Four types of question

In percentage questions, read the question carefully and decide what you are being asked to do. You may need to:

• Find a given percentage of an amount.
• Work out a percentage when given 2 amounts.
• Work backwards from a percentage increase or decrease (reverse percentages).
• Find a cumulative change.

Now try a Test Bite