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Revision:Percentages

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Percentages

1 Percentages
- 1.1 Example
2 Percentage Change
- 2.1 Example
3 Percentage Error
- 3.1 Example
4 Original value
- 4.1 Example
5 Percentage Increases and Interest
- 5.1 Example
- 5.2 Compound Interest
6 Percentage decreases
- 6.1 Example
7 Comments

Percentages

A percentage is a fraction whose denominator is 100 (the numerator of a fraction is the top term, the denominator is the bottom term). So:

$30\% = \frac{30}{100} = \frac{3}{10} = 0.3$

To change a decimal into a percentage, multiply by 100. So:

$0.3 = 0.3 \times 100 = 30\%$ .

Example

Find 25% of 10 (remember 'of' means 'times').

$\frac{25}{100} \times 10}$ (divide by 100 to convert the percentage to a decimal)

= 2.5

Percentage Change

$\displaystyle \%\ \mathsf{change} = \frac{\mathsf{new\ value} - \mathsf{original\ value}}{\mathsf{original\ value}} \times 100$ .

Example

The price of some apples is increased from 48p to 67p. By how much percent has the price increased by?

$\displaystyle \%\ \mathsf{change} = \frac{67 - 48}{48} \times 100 = 39.58\%$ .

Percentage Error

$\displaystyle \%\ \mathsf{error} = \frac{\mathsf{error}}{\mathsf{real\ value}}\times 100$ .

Example

Nicola measures the length of her textbook as 20cm. If the length is actually 17.6cm, what is the percentage error in Nicola's calculation?

$\displaystyle \%\ \mathsf{error} = \frac{20 - 17.6}{17.6}\times 100 = 13.64\%$ .

Original value

$\displaystyle \mathsf{Original\ value} = \frac{\mathsf{New\ value}}{100 + \%\ \mathsf{change}}\times 100$ .

Example

A dealer buys a stamp collection and sells it for £2700, making a 35% profit. Find the cost of the collection.

It is the original value we wish to find, so the above formula is used.

$\displaystyle \frac{2700}{100 + 35}\times 100 = \pounds 2000$ .

Percentage Increases and Interest

$\displaystyle \mathsf{New\ value} = \frac{100 + \mathsf{percentage\ increase}}{100}\times \mathsf{original\ value}$ .

Example

£500 is put in a bank where there is 6% per annum interest. Work out the amount in the bank after 1 year.

In other words, the old value is £500 and it has been increased by 6%.

Therefore:

$\displaystyle \mathsf{new\ value} = \frac{106}{100}\times 500 = \pounds 530$ .

Compound Interest

If in this example, the money was left in the bank for another year, the £530 would increase by 6%. The interest, therefore, will be higher than the previous year (6% of £530 is more than 6% of £500). Every year, if the money is left sitting in the bank account, the amount of interest paid would increase each year. This phenomenon is known as compound interest.

The simple way to work out compound interest is to multiply the money that was put in the bank by , where:

$\displaystyle n = \frac{100 + \mathsf{percentage\ increase}}{100}$ and

is the number of years the money is in the bank for, i.e:

(diagram missing)

So if the £500 had been left in the bank for 9 years, the amount would have increased to:

(diagram missing)

Percentage decreases

$\displaystyle \mathsf{New\ value} = \frac{100 - \mathsf{percentage\ decrease}}{100}\times \mathsf{original\ value}$ .

Example

At the end of 1993 there were 5000 members of a certain rare breed of animal remaining in the world. It is predicted that their number will decrease by 12% each year. How many will be left at the end of 1995?

At the end of 1994, there will be $\displaystyle \frac{100 - 12}{100}\times 5000 = 4400$ .