We use percent (hundredths) and per mille (thousandths) due to the fact that our numeral system is made up of ones, tens, tenths, hundredths, and so on. Dividing by one hundred is the same as moving the decimal mark two places to the left. You have already used this rule in many examples so far.
Rule
When converting from a percentage to a decimal number, you remove the percent sign % and move the decimal mark two places to the left.
When converting a decimal number to percent, you add a percent sign % and move the decimal mark two places to the right.
This rule is valid because an equality can be read both from left to right and from right to left.
What you need to remember about per mille is that it means thousandth, which means you have to move the decimal mark three places. It is otherwise exactly the same process as when you work with percentages. Per mille is used much less often than percentages are, so the most important thing is to know percentages. That’s also why we’ll mostly focus on percentages here.
Example 1
Here, we convert $6$ % to a decimal number and back: $$\begin{array}{llll}\hfill 6\phantom{\rule{0.17em}{0ex}}\text{\%}& =6\times \frac{1}{100}=\frac{6}{100}=0.06\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 0.06& =\frac{6}{100}=6\times \frac{1}{100}=6\phantom{\rule{0.17em}{0ex}}\text{\%}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
And here, we convert $75$ % to a decimal number and back: $$\begin{array}{llll}\hfill 75\phantom{\rule{0.17em}{0ex}}\text{\%}& =6\times \frac{75}{100}=\frac{75}{100}=0.75\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 0.75& =\frac{75}{100}=75\times \frac{1}{100}=75\phantom{\rule{0.17em}{0ex}}\text{\%}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
You can see that the calculations on the top are just the opposite operations of the calculations on the bottom, and writing things twice is unnecessary.
Example 2
$$200\phantom{\rule{0.17em}{0ex}}\text{\%}=200\times \frac{1}{100}=\frac{200}{100}=2$$ |
$200$ % of a pizza is two pizzas. That’s twice as much as one pizza, or double the amount of pizza.
Moving the decimal mark two places is the same thing you do when you switch between meters and centimeters. You move two places to the left when you convert from centimeters to meters, and two places to the right when you convert from meters to centimeters.
Example 3
You get a similar computation when you convert $6$ cm into meters by using decimal numbers. You rely on the conversion $1\phantom{\rule{0.17em}{0ex}}\text{cm}=\frac{1}{100}\phantom{\rule{0.17em}{0ex}}\text{m}$, just like $1\phantom{\rule{0.17em}{0ex}}\text{\%}=\frac{1}{100}$:
$$\begin{array}{llll}\hfill 6\phantom{\rule{0.17em}{0ex}}\text{cm}& =6\times \phantom{\rule{-0.17em}{0ex}}\left(\frac{1}{100}\phantom{\rule{0.17em}{0ex}}\text{m}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\phantom{\rule{-0.17em}{0ex}}\left(6\times \frac{1}{100}\right)\phantom{\rule{0.17em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{6}{100}\phantom{\rule{0.17em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.06\phantom{\rule{0.17em}{0ex}}\text{m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
Remember to move the decimal mark in the correct direction. If you’re not sure, use the trick that $1=\frac{100}{100}$ and use the resulting fraction. You can also use these memory aids:
Rule
Remember that cent in percent (per-cent) means hundredth, just like cent in centimeters means hundredth.
$6$ % is a small part of the whole, just like $6$ cm is a small part of 1 meter.
$60$ % is between half and the entirety of a whole, just like $60$ cm is between half a meter and a full meter.
$600\phantom{\rule{0.17em}{0ex}}\text{\%}=6$ is 6 times as much as 1, just like $600\phantom{\rule{0.17em}{0ex}}\text{cm}=6\phantom{\rule{0.17em}{0ex}}\text{m}$ is 6 times as long as one meter.
This rule can also be used when you’re looking at, for example, $7$ % and $70$ % or $9$ % and $90$ %. You can see that there is a big difference between $6$ %, $60$ % and $600$ %. It’s very important to place the decimal mark correctly! There is a huge difference between $$600$ and $$6$, so make sure you keep the direction straight!