Percentages : Expressing a Number as a Percentage of Another
In this section we learn how to express one number as a percentage of annother. For example, by the end of this section we'll have no trouble showing that: \[18 = 45\% \ \text{of} \ 40\] We start by learning the method as well as read through a worked example before working through some exercises.
Tutorial: Expressing a Number as a Percentage of Another
In the following tutorial we learn how to express one number as a percentage of another.
Method: expressing \(A\) as a percentage of \(B\)
Given two numbers \(A\) and \(B\), to express \(A\) as a percentage of \(B\) we
- Step 1: divide \(A\) by \(B\), that's \(A\div B\) or \(\frac{A}{B}\).
- Step 2: write the fraction found in step 1 as an equivalent fraction over \(100\). Using the fact that \(X\% = \frac{X}{100}\) we can then state our final answer.
Example 1
Express \(3\) as a percentage of \(5\).
Solution
Since \(5\) is a factor of \(100\) (\(100\) is a multiple of \(5\)) we don't need a calculator. We follow our two-step method:
- Step 1: we write \(3\div 5\) as a fraction: \[\frac{3}{5}\]
- Step 2: we write \(\frac{3}{5}\) as an equivalent fraction over \(100\). Using the fact that \(100 = 5\times 20\), we multiply both the numerator and the denominator by \(20\) to obtain our fraction: \[\frac{3}{5} = \frac{3\times 20}{5\times 20} = \frac{60}{100}\] Finally, since \(\frac{60}{100} = 60\%\) we can state that \(3\) is \(60\%\) of \(5\).
Example 2
Express \(18\) as a percentage of \(40\).
Solution
Since \(40\) isn't a factor of \(100\) (\(100\) isn't a multiple of \(40\)) we use a calculator for step 1. With that in mind, we follow our two-step method:
- Step 1: we start by dividing \(18\) by \(40\): \[ \frac{18}{40} = 0.45\] This tells us that \(\frac{18}{40}\) is equivalent to: \[\frac{0.45}{1}\]
- Step 2: We write \(\frac{0.45}{1}\) as an equivalent fraction over \(100\). To do this we multiply the numerator and the denominator by \(100\): \[\frac{0.45}{1} = \frac{0.45 \times 100}{1\times 100} = \frac{45}{100}\] Finally, since \(\frac{45}{100} = 45\%\) we can state that \(18\) is \(45\%\) of \(40\).
Exercise 1
Answer each of the following questions, rounding to \(1\) decimal place when appropriate.
- Express \(17\) as a percentage of \(20\)
- Express \(30\) as a percentage of \(80\).
- Express \(2\) as a percentage of \(5\)
- Express of \(25\) as a percentage of \(60\).
- What percentage of \(92\) is \(43\)?
- What percentage of \(110\) is \(21\)?
- Express \(8\) as a percentage of \(12.5\)
- Express \(35\) as a percentage of \(240\).
- Charlotte weighs \(18\)kg. Her dad weighs \(74\)kg. What percentage of her dad's weight does Charlotte weigh? Express Charlott'es dad's weight as a percentage of her's.
- Benjamin has been saving-up for a game that costs \(\$ \ 115,00\). So far he has saved \(\$ \ 90,00\). What percentage of the game's price has Benjamin managed to save so far?
- Every summer, John and his family drive \(1340\)km to reach their summer hourse. After having driven \(850\)km, John's eldest daughter Clara decides to calculate the percentage of the trip they've already travelled. What percentage does she find?
- In order to lose weight Serge decided to cut his daily calory intake to \(2200\) calories, instead of the \(2500\) calories he used to take. Express his new daily intake of calories as a percentage of the previous.
- James wins \(\$ \ 3400,00\) at the city hall lottery. He decides to give \(\$ \ 1150,00\) to his brother Jonathan to help him buy the laptop he wanted. Express the amount James gave his brother as a percentage?
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To prepare for her exams Sarah develops a study plan. Every week she studies:
- \(9\) hours of mathematics
- \(6\) hours of physics
- \(5\) hours of computer science
- \(4\) hours of chemistry
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