# Per Mille

Sometimes you come across ratios that are better described using thousandths instead of hundredths. When measuring how much alcohol is in someone’s blood to see if someone has been driving under the influence, for example, some countries use per mille. The limit set for commercial drivers in the United States, according to federal law is $0.4$ per mille. That means they need to have less than $0.4$ milliliters of pure alcohol in 1 liter of their blood to be allowed to drive.

Rule

Per mille means thousandth. The sign looks like this:

• You can always exchange the per mille sign  for $\frac{1}{1000}$

• You can always exchange $n\phantom{\rule{0.17em}{0ex}}\text{‰}$ for $\frac{a}{1000}$

Example 1

Write $\text{}0.4\text{}\phantom{\rule{0.17em}{0ex}}\text{‰}$ as a decimal number

$\begin{array}{llll}\hfill 0.4\phantom{\rule{0.17em}{0ex}}\text{‰}& =0.4×\frac{1}{1000}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{0.4}{1000}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{0.4×10}{1000×10}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{4}{10\phantom{\rule{0.17em}{0ex}}000}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.0004\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Note that you can just write $0.4$ as the numerator of the fraction. You can expand it by 10 later on.

If you take out a loan at the bank, the interest rate might be $3.5$ % per year.

Example 2

Write $\text{}3.5\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ as per mille

$\begin{array}{llll}\hfill 3.5\phantom{\rule{0.17em}{0ex}}\text{%}& =\frac{3.5}{100}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{3.5×10}{100×10}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{35}{1000}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =35\phantom{\rule{0.17em}{0ex}}\text{‰}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$