Finding a percentage
A percentage is a proportion that shows a number as parts per hundred. The symbol '%' means 'per cent'. 9% means 9 out of every 100, or \(\frac{9}{100}\).
Percentages are just one way of expressing numbers that are part of a whole. These numbers can also be written as fractionA fraction is a part of a whole, for example 1/2. or decimalA number that uses powers of 10 as place value. In the example of 0.82, the 8 represents tenths and the 2 represents hundredths.. 50% can also be written as a fraction, \(\frac{1}{2}\), or a decimal, 0.5. They are all exactly the same amount.
Knowledge of converting between decimals, fractions and percentages is required.
Calculating percentages of amounts
Percentages of amounts can be calculated by writing the percentage as a fraction or decimal and then multiplying it by the amount in question.
Example 1
Find 16% of 40.
16% is the same as \(\frac{16}{100}\).
To find 16% of 40, multiply \(\frac{16}{100}\) by 40:
\(\frac{16}{100} \times 40\)
\(= \frac{16}{100} \times\frac{40}{1}\)
\(= \frac{16 \times 40}{100 \times 1}\)
\(= \frac{640}{100} = 6.4\) (to divide by 100, bring the decimal place in by two places)
This example could also be worked out by converting the percentage to a decimal. However, this method may be easier to do if a calculator is available.
16% is the same as 0.16. To find 16% of 40, multiply 0.16 by 40:
\(0.16 \times 40 = 6.4\).
Example 2
Percentages of amounts can also be found using known facts about percentages. The most helpful of these facts is how to find 10% of an amount.
10% can be written as \(\frac{10}{100}\) because 10% means 10 out of every 100. simplifyA fraction is simplified when there are no more common factors shared by the numerator and denominator. For example, the fraction 8/10 simplifies to 4/5 by dividing the numerator and denominator by the common factor of 2. the fraction \(\frac{10}{100}\) gives \(\frac{1}{10}\) (taking out a common factorA whole number that divides into two (or more) other numbers exactly, eg 4 is a common factor of 8, 12 and 20. of 10).
This means that 10% is equivalent to dividing by 10, or finding \(\frac{1}{10}\) of the amount.
As finding 10% of a number means to divide by 10, it is common to think that to find 20% of a number you should divide by 20 etc. Remember, to find 10% of a number means dividing by 10 because 10 goes into 100 ten times. Therefore, to find 20% of a number, divide by 5 because 20 goes into 100 five times.
Once 10% of an amount is known, this can be manipulated to find other amounts such as 5% or 1%, or any amount that is helpful to answer the question.
Question
Find 27% of 80.
Start by finding 10% of 80 by dividing 80 by 10:
\(80 \div 10 = 8\)
The question asks for 27%, so double 10% to get the 20% needed:
\(8 \times 2 = 16\)
Now that 20% has been found, the remaining amount to calculate the full 27% is 7%. This can be done in lots of different ways, but one of the easiest ways is to find 1% and multiply this by 7.
1% = \(\frac{1}{100}\) so 1% can be found easily by dividing the amount by 100.
1% of 80 = \(80 \div 100 = 0.8\)
So 7% = \(0.8 \times 7 = 5.6\).
20% of 80 = 16 and 7% of 80 = 5.6
27% of 80 = \(20 \% + 7 \% = 16 + 5.6 = 21.6\)