# Percentages

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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% = 30/100 = 3/10 = 0.3
To change a decimal into a percentage, multiply by 100. So 0.3 = 0.3 × 100 = 30% .

Example:
Find 25% of 10  (remember ‘of’ means ‘times’).
25  ×  10  (divide by 100 to convert the percentage to a decimal)
100
= 2.5

Percentage Change
% change  =  new value – original value   × 100
original value

Example:
The price of some apples is increased from 48p to 67p. By how much percent has the price increased by?
% change = 67 – 48  × 100  =  39.58%
48

Percentage Error
% error =     error    × 100
real value

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?
% error = 20 – 17.6 × 100  =  13.64%
17.6

Original value

Original value =   New value       × 100
100 + %change

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.
2700     × 100   = £2000
100 + 35

Percentage Increases and Interest
New value = 100 + percentage increase × original value
100

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, new value = 106/100 × 500 = £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 nm, where n is (100 + percentage increase)/100 and m is the number of years the money is in the bank for, i.e:

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

Percentage decreases:

New value = 100 – percentage decrease × original value
100

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 (100 – 12)/100 × 5000 = 4400
At the end of 1995, there will be 88/100 × 4400 = 3872

The compound interest formula above can also be used for percentage decreases. So after 4 years, the number of animals left would be:

5000 x [(100-12)/100]4  = 2998