Converting Percentages and Fractions

A fraction \(\frac{m}{n}\) represents the quantity of \(m\) divided by \(n.\) For example, a half can be represented by \(\frac{1}{2}.\) When we have three halves or three divided into two pieces, we eventually have \(\frac{3}{2}.\) A percentage (%) is essentially also a fraction with the denominator 100. Namely, a percentage is the quantity by (per) the hundred (centage).

For example, 2%, read "percent", is two every hundred. When we have 100 liters of water, 2% is 2 liters. If we have 200 liters of water, then 2% is 4 liters.

Converting a percentage into a fraction can be done by dividing by 100, following the definition of a percentage. Converting a fraction into a percentage can be done by multiplying by 100, which is the reverse procedure of the former conversion.

In order to change a fraction into a decimal, divide the numerator by the denominator.

Convert \(\frac{1}{5}\) to decimal.

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We have \[ 1 \div 5 = 0.2. \ _\square\]

Convert \(\frac{1}{6}\) to decimal.

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We have \[1 \div 6 = 0.16666666 \ldots. \] When we encounter decimals with infinite digits, we can simply put a horizontal bar over the string of digits that repeat: \[1 \div 6 = 0.1\overline{6}. \ _\square\]

If we'd like to convert a fraction to a percent, we multiply the fraction by \(100\) and then perform the division.

Convert \(\frac{1}{5}\) to a percentage.

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We have \[\frac{1}{5} \times 100 \text{%}= \frac{100}{5} \text{%} = 20 \text{%}. \ _\square\]

If the number of digits in the fractional part of a decimal is \(n,\) then the corresponding fraction is the fractional part divided by \(10^n.\)

Convert 0.837 to a fraction.

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We have \[0.837 = \frac{837}{10^3} = \frac{837}{1000}. \ _\square\]

To obtain a percent, shift the decimal point two places to the right and add the percent sign.

Convert 0.1024 to a percentage.

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We have \[0.1024 = 10.24 \text{%}. \ _\square\]

To obtain a fraction, divide the percentage by \(100,\) and reduce. In short, \(x\% =\cfrac { x }{ 100 }. \)

Convert 6% to a fraction.

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We have \[ 6 \text{%} = \frac{6}{100} = \frac{3}{50}. \ _\square\]

In order to get a decimal, shift the decimal point two places to the left and remove the percent sign.

Convert 0.08% to a decimal.

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We have \[ 0.08 \% = 0.0008 .\ _\square\]

What is \({ (100\%) }^{ 2 }\) in percentage?

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We can solve this problem in a simpler way by transforming the percentage into a fractional value:

\[100\% = \frac{100}{100}.\]

When we raise a number to a power, we essentially multiply it by itself. In this case we need to multiply \(\frac{100}{100}\) two times (indicated by the exponent).

Mathematically,

\[\frac{100}{100}\times \frac{100}{100} = \frac{10000}{10000}.\]

If we simplify our answer \(\frac{10000}{10000},\) we get \(\frac{100}{100}\) again, which as we know is 100%. \(_\square\)

Note: Yes, we can say \(100\% = 1,\) but for mathematical presentation and to make it easier to understand the concept of squaring percentages, I used \(100\% = \frac{100}{100}.\)

So far, all we have been doing is problems trying to strengthen your mastery of conversion between fractions and percentages and vice versa. But how do these concepts apply to the real world? Well, when you get a test back from one of your teachers, typically there will be a grade at a top. There will most likely be a fraction or percentage at the top of your paper.

Frank got back a math test and it said he got 19 questions right out of 20. What percentage of the questions did he answer correctly?

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First, understand that the fraction is \(\frac {19}{20}\). Then multiply both the numerator and denominator of this fraction by 5, since \(20 \times 5 = 100\). You would get \(\frac {95}{100}\). Thus, \(\frac {19}{20}\) expressed as a percentage is 95%. \(_\square\)

Clara just got back the results of the dreaded 70 question biology exam. She answered 85% of the questions correctly. How many questions did she answer correctly?

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First, you should set up a proportion where \(\frac {85}{100} = \frac {x}{70}\). Then cross multiply and set up the equation \(100x = 5950\). To solve for \(x\), divide \(5950\) by \(100\), giving us the number \(59.5\). So \(x = 59.5\). To answer the question, Clara got \(59.5\) questions correct out of a total of \(70\) questions. This can be expressed as the fraction \(\frac {59.5}{70}\). \(_\square\)