Percentages, Parts per Million, and Parts per Billion

In our daily lives, we read and hear about 85 percent of voters supporting a certain bill, or drinking water containing a certain parts per million or parts per billion of toxic substances. What do these numbers actually mean and how are they calculated? In this section, we will not only become more familiar with their meaning, we will also see that the calculations involved are very similar. Let’s start with percentages.

Percentages

Percent is defined as a fraction of the whole expressed as parts per hundred.

In a newspaper we read that in a certain city, 75 percent (75%) of the residents support the building of a park alongside the city’s river. Let’s say that 3800 people were polled on this issue and 2850 of the residents supported this issue. To perform the calculation we first take the number of people who support the issue and divide that number by the total number of people polled.

\(\displaystyle \frac{2850 (in favor)}{3800 (total polled)}\;=\;0.75\;\)
 
The 0.75 represents a fraction of all people polled who favor the park construction. Normally, we think of fractions as rational numbers (numbers expressed as ratios) that might be expressed, for example, as 1⁄2 (a 1 to 2 ratio), 1⁄4 (a 1 to 4 ratio), or 1⁄8 (a 1 to 8 ratio). These same fractions can also be expressed as 0.5, 0.25, and 0.125, that is, as decimals. Fractions can easily be converted into decimals by dividing the bottom number (the denominator) into the upper number (the numerator).

Continuing our discussion, we know the next step of multiplying this fraction by 100 to calculate the percentage.

0.75 x 100 = 75%

But why do we do this? Why not just use the fraction by itself? Why multiply by 100, let alone 10 or 1000? By multiplying by 100, we are saying that out of every 100 people polled, 75 people support the park. We could easily have multiplied the fraction by 10 or 1000. Note that traditionally we use groupings of 100 in most of our everyday analysis. If we had multiplied by 10, the analysis would state that 6.5 people out of every 10 supported the issue. If we multiplied by 1000, the result would read that for every 1000 people questioned, 650 were in favor. Traditionally, we have settled on 100, which is why it is called a percent, that is parts per 100.

Note that the calculated percentages add up to 100% and the fractions add to 1. This is true for all calculations of this sort.

Parts per Million (ppm) and Parts per Billion (ppb)

Let’s now consider situations where multiplying fractions by 100 isn’t adequate because the resultant percentages are much too small. Suppose we are doing an analysis of drinking water and find that a 1000 gram sample of the water contains 0.0002 grams of lead ion. To use these numbers in a percentage would yield the following results:

\(\displaystyle \frac{0.0002\;g\;lead}{1000\;g\;water}\times 100\;=\;0.00002% lead in water\;\)

To better handle such small numbers, instead of multiplying the fraction by 100, we use 1,000,000 (one million) and by doing, the result is no longer called percentage or parts per hundred, but rather parts per million, ppm. By multiplying the small fraction by 1,000,000 (10⁶) a more manageable number is produced:

\(\displaystyle \frac{0.0002\;g\;lead}{1000\;g\;water}\times 10^6\;=\;0.2\; ppm\;\)

When dealing with chemical pollutants and toxic microorganisms, due to their small concentrations, we normally use parts per million, ppm or even parts per billion, ppb. Multiplying the fraction of lead by 1 billion (1,000,000,000) we obtain:

\(\displaystyle \frac{0.0002\;g\;lead}{1000\;g\;water}\times 10^9\;=\;200\; ppb\;\)

Click Here for Percentage and Parts Per Million Worksheet

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