# Percentages

## What are percentages?

A *percentage* is a decimal fraction with denominator 100.

"*Per cent*" or "*per centum*" literally means "per hundred" ("in each hundred") in Latin.
Percent (per cent) indicates parts per hundred.

A percentage is usually given as the numerator followed by a "%" sign,
i.e. the *percent sign*: 25% pronounced as "twenty-five percent":

$$\frac{25}{100}=0.25=25\mathrm{\%}$$

The numerator can be an integer or a fraction itself:

$$12.15\mathrm{\%}=0.1215$$

Half of a cake is the same as 50% of a cake or 4/7 of the animals in the shelter are a cat is (approximately) the same as 57.14% of the animals in the shelter are a cat.

$$\frac{1}{2}=\frac{\frac{1}{2}\times 100}{100}=\frac{50}{100}=50\mathrm{\%}$$

$$\begin{array}{rl}\frac{4}{7}=& \frac{\frac{4}{7}\times 100}{100}=\frac{\frac{400}{7}}{100}\\ \phantom{\frac{4}{7}}=& \frac{400}{7}\mathrm{\%}\approx 57.14\mathrm{\%}\end{array}$$

The numerator of a percentage is usually expressed in decimal notation. An expression like 400/7 % is unusual. A fraction with a denominator whose prime factors are only twos and/or fives can be precisely expressed as a decimal fraction and thus can be precisely expressed as a percentage with a numerator being a terminating decimal expansion. Other fractions expressed as percentages have numerators that are non-terminating decimal expansions, which in practice are approximated by decimal fractions.

$$\frac{1}{2}=0.5=\frac{0.5\times 100}{100}=\frac{50}{100}=50\mathrm{\%}$$

$$\frac{1}{8}=0.125=12.5\mathrm{\%}$$

$$\frac{1}{3}=0.\overline{)3}=33.\overline{)3}\mathrm{\%}\approx 33\mathrm{\%}$$

$$\begin{array}{rl}\frac{4}{7}=& 0.\overline{)571428}\\ \phantom{\frac{4}{7}}=& 57.1428\mathrm{\dots}\mathrm{\%}\approx 57.14\mathrm{\%}\end{array}$$

0. | 5 | 7 | 1 | 4 | 2 | 8 | ||

7 | ) | 4. | 0 | 0 | 0 | 0 | 0 | |

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Learn the next percentages by heart:

fraction | decimal | percentage |
---|---|---|

$\frac{1}{2}$ | 0.5 | 50% |

$\frac{1}{3}$ | ≈ 33.33% | |

$\frac{1}{4}$ | 0.25 | 25% |

$\frac{1}{5}$ | 0.2 | 20% |

$\frac{1}{8}$ | 0.125 | 12.5% |

$\frac{1}{10}$ | 0.1 | 10% |

$\frac{1}{20}$ | 0.05 | 5% |

$\frac{1}{25}$ | 0.04 | 4% |

$\frac{1}{40}$ | 0.025 | 2.5% |

$\frac{1}{50}$ | 0.02 | 2% |

Now reduce 40%, 60%, 75%, 80%, 100% and 225% to lowest terms. What percentages are approximately equal to $\frac{1}{6}$ and $\frac{1}{7}$?

## Percentages in word problems

Percentages are dimensionless ratios. A ratio is a quantity *relative* to an other quantity, as we have seen in the previous part.
We can use a percentage to calculate an *absolute* quantity (the actual measurable or countable amount) if we, for example, know the total absolute value.
For example, with a given unemployment rate of 5% and a total number of potential workers of 1 million, we can calculate the actual number of unemployed people.

We say "five percent of one million equals fifty thousand unemployed persons". As we've seen earlier with fractions in general, the word "of" needs to be interpreted as "times". "Five percent of one million" means 5% × 1 000 000.

Seventy percent of 200:

$$\begin{array}{rl}70\mathrm{\%}\times 200=& \frac{70\times 200}{100}\\ \phantom{70\mathrm{\%}\times 200}=& 70\times 2=140\end{array}$$

This is actually solving a proportion, as we covered in the previous part, i.e.,
70 to 100 is as `x` to 200.
We need to convert 70 per 100 to a fraction per 200:

$$\begin{array}{rl}\frac{70}{100}=& \frac{x}{200}\\ 100\times x=& 70\times 200\\ x=& \frac{70\times 200}{100}=140\end{array}$$

How to calculate the percentage?
"What percentage of 200 is 140?" or "how many percent is 140 of 200?".
We can use a little algebra and fill in a *variable* for percentage or total in the initial equation:

$$\begin{array}{rl}p\times 200=& 140\\ \frac{p\times 200}{200}=& \frac{140}{200}\\ p=& \frac{70}{100}=70\mathrm{\%}\end{array}$$

Or:

$$\begin{array}{rl}\frac{p}{100}=& \frac{140}{200}\\ 200\times p=& 140\times 100\\ p=& \frac{140\times 100}{200}=70\end{array}$$

And now we calculate the total, i.e., "70% of how much is 140?"

$$\begin{array}{rl}70\mathrm{\%}\times t=& 140\\ t=& \frac{140}{70\mathrm{\%}}\\ t=& \frac{140\times 100}{70}=200\end{array}$$

Or: 70 to 100 is as 140 to the total, that is, 200.

$$\begin{array}{rl}\frac{70}{100}=& \frac{140}{t}\\ 70\times t=& 140\times 100\\ t=& \frac{140\times 100}{70}=200\end{array}$$

This all is basically nothing more than division being the opposite of multiplication, as we have seen before. Only now we convert to per 100 instead of per 1.

$\frac{140}{200}=70\mathrm{\%}$ because $70\mathrm{\%}\times 200=140$.

Can you see why the "general formula" of a percentage is sometime presented as:

$$\mathrm{percentage}=\frac{\mathrm{part}}{\mathrm{total}}\times 100$$

When we calculate with percentages we mostly calculate a part of a whole, i.e., of some total. The total corresponds to 100%. The difficulty with percentages (and fractions in general) is often to find the right total. Furthermore, in word problems often the need to add or subtract a percentage from a total is concealed in the posing of the problem.

Another point of special interest is the addition and subtraction of percentages. Percentages can only be added or subtracted correctly if they are relative to the same total.

### Examples

A constructor offers to install a new bathroom for a price of €2000 before 12% value-added tax. What is the total price after tax?

Total price:

$$112\mathrm{\%}\times \mathrm{\u20ac}2000=\mathrm{\u20ac}2240$$

Or:

$$\begin{array}{rl}\frac{112}{100}=& \frac{p}{2000}\\ p=& \frac{112\times 2000}{100}\\ p=& \frac{112\times 20\times 100}{100}=2240\end{array}$$

1 | 1 | 2 | |

2 | 0 | ||

0 | |||

2 | 2 | 4 | 0 |

2 | 2 | 4 | 0 |

The 12% tax is of €2000. This €2000 is the total, the 100%. So, 112% (100% + 12%) of €2000 is the price after tax.

A store offers a discount on cakes with a regular price of $11. A customer pays $9.35. How much is the relative discount expressed as a percentage?

The discount is the next ratio expressed as a percentage:

$$\frac{\mathrm{\$}1.65}{\mathrm{\$}11}=15\mathrm{\%}$$

$$\frac{1.65}{11}=\frac{165}{11\times 100}=\frac{11\times 15}{11\times 100}=\frac{15}{100}$$

1 | 5 | |||

11 | ) | 1 | 6 | 5 |

1 | 1 | ↓ | ||

5 | 5 | |||

5 | 5 | |||

0 |

Or:

$$\begin{array}{rl}\frac{1.65}{11}=& \frac{p}{100}\\ p=& \frac{100\times 1.65}{11}\\ p=& \frac{165}{11}=15\end{array}$$

The absolute discount is $1.65 ($11 − $9.35). The relative discount expressed as a percentage is as the absolute discount relative to the regular price of $11 (the total).

A salesman earns a base salary of $150.00 per week with an additional relative commission on everything he sells. Last week this salesman sold $6050.00 worth of items, and his total pay was $876.00. What is his commission expressed as a percentage?

His absolute commission was $726 ($876 − $150). The relative commission is:

$$\frac{\mathrm{\$}726}{\mathrm{\$}6050}=12\mathrm{\%}$$

$$\frac{726}{6050}=\frac{6\times 121}{50\times 121}=\frac{12}{100}$$

0 | . | 1 | 2 | ||||

6050 | ) | 7 | 2 | 6 | . | 0 | 0 |

0 | ↓ | ||||||

7 | 2 | 6 | 0 | ||||

6 | 0 | 5 | 0 | ↓ | |||

1 | 2 | 1 | 0 | 0 | |||

1 | 2 | 1 | 0 | 0 | |||

0 |

Or:

$$\begin{array}{rl}\frac{726}{6050}=& \frac{p}{100}\\ p=& \frac{100\times 726}{6050}\\ p=& 12\end{array}$$

## Compound percentages

A compounding percentage is a percentage of a percentage or percentages applied sequentially.
Compounding percentages cannot be calculated by simply adding up the percentage values and apply this "sum percentage" to the total.
The problem is that different percentages refer to different totals. A 5% increase in price followed by a 5% decrease in price *will* change
the final price. An extra 40% discount on prices that are already 60% off does not mean that you will get the product for free.

Figure 1 shows a discount of 40% on prices already discounted 60%. The 60% refers to the original price and the 40% refers to the already discounted price. If both percentages were calculated of the original price, than indeed the final price would be zero and the total discount would be 100% = 1. The correct calculation of the final price is however: 60% of 40% of the original price.

$$\begin{array}{l}\mathrm{final\; price}=60\mathrm{\%}\times 40\mathrm{\%}\times \mathrm{initial\; price}\\ \phantom{\mathrm{final\; price}}=\frac{60}{100}\times \frac{40}{100}\times \mathrm{initial\; price}\\ \phantom{\mathrm{final\; price}}=\frac{2400}{10000}\times \mathrm{initial\; price}\\ \phantom{\mathrm{final\; price}}=24\mathrm{\%}\times \mathrm{initial\; price}\end{array}$$

$$\begin{array}{l}\mathrm{total\; discount}=\mathrm{initial\; price}-\mathrm{final\; price}\\ \phantom{\mathrm{total\; discount}}=100\mathrm{\%}-24\mathrm{\%}=76\mathrm{\%}\end{array}$$

That is 76% of the initial price. Percentages can be subtracted because they refer to the same total (the initial price).

Suppose a price rises 5% and then drops 10%. Then the new price will be 94.5% of the original price:

$$\begin{array}{l}\mathrm{new\; price}=90\mathrm{\%}\times 105\mathrm{\%}\times \mathrm{initial\; price}\\ \phantom{\mathrm{new\; price}}=\frac{90}{100}\times \frac{105}{100}\times \mathrm{initial\; price}\\ \phantom{\mathrm{new\; price}}=\frac{9450}{10000}\times \mathrm{initial\; price}\\ \phantom{\mathrm{new\; price}}=94.5\mathrm{\%}\times \mathrm{initial\; price}\end{array}$$

In total, the price drops 100% – 94.5% = 5.5% of the initial price, from the original price.

Note that the total price decrease cannot be calculated as 10% × 5% × initial price.
This calculates 10% of the initial absolute rise, while 10% refers to the *total* increased price.
The price drop from the interim high (the price after the 5% rise) is 10% × 105% × initial price = 10.5% × initial price.

### Percentage points

Suppose an interest rate rises from 5% per annum to 7% per annum. It is popular, yet fallacious, to say that in this case the interest rate increases by 2%. One takes the "sum percentage", 7% – 5% = 2%, and call that the increase.

However, if we take absolute interests and calculate the interest rise in percentages:

$$\begin{array}{l}\frac{7\mathrm{\%}\times \mathrm{savings}}{5\mathrm{\%}\times \mathrm{savings}}=\frac{x}{100}\\ x=\frac{7\mathrm{\%}}{5\mathrm{\%}}\times 100\\ \phantom{x}=140\end{array}$$

So, the interest rises 40% and not 2%! A rise of 2% of the initial rate of 5% results in a new interest rate of 5.1%, instead of 7%:

$$\begin{array}{l}102\mathrm{\%}\times 5\mathrm{\%}=\frac{102}{100}\times \frac{5}{100}\\ \phantom{102\mathrm{\%}\times 5\mathrm{\%}}=\frac{510}{10000}=5.1\mathrm{\%}\end{array}$$

So, a rise from 5% per annum to 7% per annum is a rise of 40%:

$$\begin{array}{rl}p\times 5\mathrm{\%}=& 7\mathrm{\%}\\ p=& \frac{7\mathrm{\%}}{5\mathrm{\%}}=\frac{140}{100}=140\mathrm{\%}\end{array}$$

To avoid confusion, we say that an increase from 5% to 7% is an increase of 40%, or an increase of 2
*percentage points*, rather than 2%.
A percentage point or *percent point* (pp) is a unit to express the arithmetic difference between two percentages.