Converting between Fractions, Decimals, and Percentages

A rational number is any number than can be written as a ratio of two integers (a fraction), such as $\frac{4}{5}$ or $\frac{8}{25}$. Percentages (such as 32%) and decimals (such as .32) are other forms in which we can write rational numbers, and it is often helpful to re-express them in different ways. For example, we might note that $\frac{8}{25} = .32 = 32\%$ are all different ways of writing the same value.

To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert $\frac{1}{4}$ to a decimal, we divide 1 by 4 to get the quotient 0.25.
$$\frac{1}{4}=0.25$$

What is the decimal representation of $\frac{1}{6}?$

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ANSWER

Performing long division when $1$ is divided by $6$, the quotient doesn't have a terminating point in that the 6 goes on forever. So, to represent a repeating number, we use a hyphen on the top: $$\frac{1}{6}=0.1\overline{6}.$$

What is the decimal representation of $\frac{1}{5}?$

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ANSWER

Performing long division when 1 is divided by 5, we get the quotient to be 0.2. Thus,

$$\frac{1}{5}=0.2.$$

To write an integer as a fraction, make the denominator one. For example, the number $9$ can be written in fractional form as $\frac{9}{1}$.

To write a terminating decimal as a fraction, write the decimal number into an equivalent fraction by making the numerator the number itself and the denominator 1. For example, if the number is $0.6$, it becomes $\frac{0.6}{1}$. After that, multiply both the numerator and the denominator with a number that takes the decimal place of the numerator into the last place. So multiply by 10 $\left(\frac{0.6\times 10}{1 \times 10}\right)$ to get $\frac{6}{10}$. Simplify to get $\frac{3}{5}$.

What is the fractional representation of $0.623?$

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ANSWER

We have

$$0.623=\frac{0.623}{1}.$$

Multiply both the numerator and the denominator by 1000, to move the decimal point of the numerator to the zero's place:

$$\frac{0.623\times1000}{1\times 1000}=\frac{623}{1000},$$

which can't be simplified any further. $_\square$

To write a repeating decimal, such as 0.343434..., as a fraction, we can begin by setting it equal to a variable: $$0.\overline{34} = x.$$

Our decimal repeats after the hundredths place, so we can write another equation, multiplying both sides of our original equation by 100: $$34.\overline{34}... = 100x.$$

Subtracting the first equation from the second, we get $34 = 99x,$ or $x = \frac{34}{99}.$

What is the fractional representation of $0.\overline{8}?$

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ANSWER

We have

$$0.\overline{8}=x.$$

Multiply both sides of the equation by 10:

$$8\overline{8}=10x.$$

Subtracting the first equation from the second, we have $8 = 9x$ and $x=\frac{8}{9}.$

You can read about other methods for converting repeating decimals to fractions here.

In everyday life, we often use percentages to express the amount of something relative to the whole. A percent is a ratio expressed as a fraction of 100, so the whole is equal to "100%". In math and science, however, decimals are more frequently used to express portions. When using decimals, we consider "1" as the whole. Therefore we have 1=100%, and we come to a formula that gives the relation between decimals and percentages:

$$\begin{aligned} a&=100\times a\text{ \%},\\ b\text{ \%}&=b\times0.01, \end{aligned}$$

where $0\leq a\leq1$ and $0\leq b\leq100$. Thus, the percentage is expressed as a number 100 times larger than the decimal. Now let's take a look at a few examples.

Convert 75% into decimals.

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Using the formula above, we have

$$75\text{ \%}=75\times0.01=0.75.\ _\square$$

How can we express 'half' using decimals? What is this in percentage?

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When we use decimals, we consider "1" as whole. Therefore "0.5" would be equal to half.

When using percentages, we consider "100%" as whole. Hence "50%" would be equal to half.

Using the conversion formula above, we can verify this as shown below:

$$0.5=100\times0.5\text{ \%}=50\text{ \%}.\ _\square$$