Percentage
Percentages are hard. Even though they are a concept that most people think they understand, there are many different types of problem, which on a surface level, appear to be similar, but actually require a wholly different method. In the section, I will be going through all of the important aspects of working with percentages, starting with what a percentage is and going on to look at reverse percentages and compound interest. Below is a list of sub-topics which we will work with here.
- What is a percentage?
- Converting between a percentage and other similar forms.
- Finding a percentage of a quantity.
- Increasing a quantity by a percentage
- Decreasing a quantity by a percentage
- Expressing one quantity as a percentage of another
- Repeated increases and decreases over time
- Undoing a percentage increase or decrease
- Summary of question types
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Worked example #1: What is a percentage?
- Notes
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Which of the following fractions could be described as a percentage?
- \[0.5\]
- \[\frac{40}{100}\]
- \[50\%\]
- \[\frac{3}{10}\]
- \[\frac{546}{100}\]
- \[750\%\]
A percentage is a fraction with a denominator of 100. The only thing that is weird about percentages is the slightly peculiar notation.
The fraction $\frac{42}{100}$ can be written as $42\%$, but there is no conceptual difference between the two. Both of them are percentages.
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- \[0.5\] This is not a percentage, as the denominator of the fraction is $2$, not $100$.
- \[\frac{40}{100}\] This is a percentage, as the denominator is $100$, so it can be written using the notation $40\%$.
- \[50\%\] This is a percentage, as it is described with the percentage notation.
- \[\frac{3}{10}\] Not a percentage, as the denominator is not $100$.
- \[\frac{546}{100}\] This is a percentage, as the denominator is $100$.
- \[750%\]
This is also a
bigger than $1$
percentage.
- $0.5$ is not a percentage, though as we will discover in the next example, it can be written as a percentage.
- This is a percentage $\frac{40}{100}$ is $40\%$.
- $50\%$ is a percentage.
- $\frac{3}{10}$ is not a percentage, though it can, like any fraction, be converted to one.
- Do not fall into the trap of thinking that you cannot have more than 100% of something. There are some things where you cannot have more than 100% of a thing, like the percentage of your effort in any circumstance, but there are many situations where you can have more than $100\%$.
- Another percentage which is bigger than $100\%$
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- No
- Yes
- Yes
- No
- Yes
- Yes
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Worked example #2: Convert a decimal to a percentage
- Notes
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Convert the following decimals to percentages.
- $0.9$
- $0.46$
- $1.03$
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- \[\begin{aligned}[t] &0.9 \times 100 = 90\\ &0.9 = 90\%\\ \hline \end{aligned}\]
- \[\begin{aligned}[t] &0.46 \times 100 = 46\\ &0.46 =46\%\\ \hline \end{aligned}\]
- \[\begin{aligned}[t] &1.03 \times 100 = 103\\ &1.03 = 103\%\\ \hline \end{aligned}\]
To convert a decimal number to a percentage, you must multiply by $100$. Nothing else. That is all!
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- $90\%$
- $46\%$
- $103\%$
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Worked example #3: Convert a terminating decimal fraction to a percentage
- Notes
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Convert $\frac{3}{5}$ to a percentage.
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\[\begin{aligned}[t] &\frac{3}{5} = \frac{60}{100} = 60\%\\ \hline \end{aligned}\] \[\begin{aligned}[t] &\frac{3}{5} = 0.6\\ &0.6 \times 100 = 60\%\\ \hline \end{aligned}\]
When you have a simple fraction, which is easy to turn into an equivalent fraction over $100$, then that is the most straight forward method to use. This is the first method on the left.
It is also perfectly okay to turn the fraction into a decimal and then multiply by $100$ to turn it into a percentage. This is the second method on the left.
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\[60\%\]
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Worked example #4: Convert a recurring decimal fraction to a percentage
- Notes
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Convert $\frac{3}{7}$ to a percentage.
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\[\begin{array}{l} \frac{3}{7} = 0.\dot 42857\dot 1\\ 0.\dot 42857\dot 1\,\, \times 100 = 42.8571\% \ (4 \text{ d.p.}) \end{array}\]
This fraction is a little harder to turn into a decimal, as sevenths make recurring decimals.
You can use a bus shelter division to work out the decimal or use a calculator. You should be able to this both ways. Certainly get used to using your calculator, but also remember that you will not always have one to hand, so it is important to be able to do the division by hand.
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\[42.9\% \ (1\text{ d.p.})\]
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Worked example #5: Converting a percentage to a decimal
- Notes
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Convert the following percentages to decimal notation:
- \[60\%\]
- \[37\%\]
- \[105\%\]
- \[283\%\]
- \[0.4\%\]
- \[0.0032\%\]
We have learned in prior examples that to turn a decimal into a percentage, we must multiply by $100$. To reverse the process, we must divide by $100$.
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- \[60 \div 100 = 0.6\]
- \[37 \div 100 = 0.37\]
- \[105 \div 100 = 1.05\]
- \[283 \div 100 = 2.83\]
- \[0.4 \div 100 = 0.004\]
- \[0.0032 \div 100 = 0.000032\]
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- \[60\% = 0.6\]
- \[37\% = 0.37\]
- \[105\% = 1.05\]
- \[283\% = 2.83\]
- \[0.4\% = 0.004\]
- \[0.0032\% = 0.000032\]
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Exercise #1: Convert decimals to percentages
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Convert each of the following decimals to percentages.
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1.\[0.4\]
\[40\%\] -
2.\[0.6\]\[60\%\]
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3.\[0.8\]\[80\%\]
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4.\[0.1\]\[10\%\]
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5.\[0.3\]\[30\%\]
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6.\[0.7\]\[70\%\]
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7.\[0.2\]\[20\%\]
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8.\[0.5\]\[50\%\]
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9.\[0.14\]\[14\%\]
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10.\[0.03\]\[3\%\]
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Exercise #2: Convert fractions to percentages
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1.\[\frac{1}{2}\]\[50\%\]
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2.\[\frac{1}{4}\]\[25\%\]
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3.\[\frac{3}{4}\]\[75\%\]
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4.\[\frac{1}{10}\]\[10\%\]
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5.\[\frac{3}{10}\]\[30\%\]
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6.\[\frac{7}{10}\]\[70\%\]
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7.\[\frac{9}{10}\]\[90\%\]
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8.\[\frac{2}{5}\]\[40\%\]
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9.\[\frac{2}{3}\]\[66.7\% \ (1 \ d.p.)\]
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10.\[\frac{1}{8}\]\[12.5\%\]
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Exercise #3: Convert percentages to decimals
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1.\[80\%\]\[0.8\]
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2.\[63\%\]\[0.63\]
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3.\[18\%\]\[0.128\]
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4.\[36\%\]\[0.36\]
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5.\[68\%\]\[0.68\]
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6.\[31\%\]\[0.31\]
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7.\[8\%\]\[0.08\]
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8.\[1\%\]\[0.01\]
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9.\[1.4\%\]\[0.014\]
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10.\[0.56\%\]\[0.0056\]
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