## Fractions are Hard!

### 3.5 Percentages

In the first century B.C.E., Roman Emperor Augustus levied for the first time a tax of one part per hundred on the proceeds of all goods sold at markets and auctions in ancient Rome. From the Latin phrase *per centum *meaning “by the hundred” came the term *percent*.

A percent is just a fraction with denominator one hundred (the number of pies per 100 boys). For example, \(\dfrac{1}{2}\) written as a fraction with denominator one hundred is

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\(\dfrac{1}{2}=\dfrac{1\times 50}{2\times 50}=\dfrac{50}{100}\).

We say that \(\dfrac{1}{2}\) equals 50 percent, and write

\(\dfrac{1}{2}=50\%\).

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Levying a tax of 1 dollar for every 100 dollars results in \(\dfrac{1}{100}\), one one-hundredth, of all cash exchanged being given to the government. This is tax rate of \(1\%\).

The symbol \(\%\) developed in Italy during the 1500s. Clerks started shortening *per cento *to P 00, which then eventually became \(%\).

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Working backwards is actually easier. For example, 80% is a fraction with denominator 100. We have

\(80\%=\dfrac{80}{100}\)

which is equivalent to the fraction \(\dfrac{4}{5}\).

Some more examples:

\(\dfrac{3}{10}=\dfrac{30}{100}=100\%\)

\(2 = \dfrac{200}{100}=200\%\)

\(0.632=\dfrac{0.632}{1}=\dfrac{0.632\times 100}{1 \times 100}=\dfrac{63.2}{100}=63.2\%\)

Question: Nervous Nelly wants a rule for understanding percentages. Someone once told her that to write a number \(x\) as a percentage, just multiply that number by 100 and write a \(\%\) sign after the result.
For example:
\(\dfrac{1}{2}\times 100 = 50\) and indeed \(\dfrac{1}{2}\) is \(50\%\).
\(0.632\times 100 = 63.2\) and indeed \(0.632\) is \(63.2\%\).
\(\dfrac{3}{7}\times 100 = \dfrac{300}{7}=42\dfrac{6}{7}\) and indeed \(\dfrac{3}{27}\) is \(42\dfrac{6}{7}\%\).
One day, maybe in the distant future, Nervous Nelly will ask why this rule works. What would you say to her? How would you explain why this rule is true? (And did she even need to memorize this as rule in the first place?) |

And backwards

\(53\% = \dfrac{53}{100}\)

\(25\% = \dfrac{25}{100}=\dfrac{1}{4}\)

\(150\% = \dfrac{150}{100}=1\dfrac{1}{2}\)

\(1000\% = \dfrac{1000}{100}=10\)

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Question: The price of a Nice-and-Spicy Soup Cup yesterday was $1.50. It’s price went up to $1.80 overnight. What percentage increase is that? |

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Question: The ancient Romans also spoke of parts per thousand. Today this is called, if it is ever used, per millage with the symbol o/oo.
For example, 467 o/oo equals \(\dfrac{467}{1000}\),
and one tenth, which is \(\dfrac{1}{10} = \dfrac{100}{1000}\), equals \(100\) o/oo.
Convert each of these per millages into fractions
a) 40 o/oo b) 500 o/oo c) 6000 o/oo d) 2 o/oo
Convert each of the following fractions into per millages:
a) \(\dfrac{1}{100}\) b) \(\dfrac{2}{5}\) c) \(30\%\) d) \(1\) e) \(0\) |

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