Expressing one number as a percentage of another number
Writing one number as a percentage of another number is a common, everyday necessity. For example, it's used for calculating percentage scores on tests or working out costs on a project.
It requires knowledge of equivalent fractionsTwo fractions are equivalent when they are worth the same amount but written differently. 3/5 is equivalent to 6/10 as the numerator and denominator have both been multiplied by 2..
Example
A maths test is out of 20 marks. A student takes this test and scores 18 marks. What percentage score did the student get?
To answer this question, first write the score as a fraction. 18 marks out of 20 can be shown as the fraction \(\frac{18}{20}\) (18 out of 20).
Percentages are 'per hundred', so to get a percentage answer the denominatorThe bottom part of a fraction. For ⅝, the denominator is 8, which represents 'eighths'. must be 100.
Use equivalent fractions to turn \(\frac{18}{20}\) into a fraction with 100 as the denominator.
\(\frac{18}{20} = \frac{90}{100}\) (multiply numeratorThe top part of a fraction. For ⅝ , the numerator is 5. and denominator by 5)
\(\frac{90}{100} = 90 \%\) (90 out of 100), so 18 out of 20 marks is 90% of the marks.
If using a calculator, you can calculate one number as a percentage of another by dividing the numbers and multiplying by 100.
Example
Find 23 as a percentage of 80.
Calculate \(23 \div 80 = 0.2875\).
\(0.2875 \times 100 = 28.75 \%\).
Question
A snack that weighs 40 g has 18 g of carbohydrates, 12 g of fat and 10 g of protein. What percentage of carbohydrates does the snack contain?
Carbohydrates are worth 18 g out of 40 g, which can be shown as \(\frac{18}{40}\).
To find \(\frac{18}{40}\) as a percentage, the denominator must be 100. This fraction has a denominator that will not easily multiply into 100, so a mid-step denominator must be made by finding 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 40 and 100. 20 is a common factor of 40 and 100 (\(20 \times 2 = 40\) and \(20 \times 5 = 100\)) so write the fraction with a denominator of 20.
\(\frac{18}{40} = \frac{9}{20}\) (divide numerator and denominator by 2) = \(\frac{45}{100}\) (multiply numerator and denominator by 5)
So 18 g out of 40 g is 45%.
Alternatively, if using a calculator, \(\frac{18}{40}\) means \(18 \div 40\) which is 0.45 as a decimal. Convert this to a percentage by multiplying it by 100:
\(0.45 \times 100 = 45 \%\)