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How to convert fractions to decimals and vice versa

Tuesday, December 27th, 2022

Fractions and Decimals

Fractions and decimals are two forms used to represent parts of whole quantities. Usually, the context determines which form works best. For example, it would be strange if the price of a pizza box reads

\$11 \frac{2}{5}

instead of $11.40. On the other hand, it wouldn't make sense to say there are 2.75 pizzas left over from last night’s party instead of 2 and 3/4 of pizza.

A Pizza box with the price indicated as eleven and two-fifths dollars

When solving algebraic equations, geometry, etc. you’ll encounter both fractions and decimals in the mix, and knowing how to convert between the two forms is vital. This guide will teach you how to turn a decimal into a fraction and vice versa step-by-step.

How to convert a decimal into a fraction?

Understanding place value in decimals gives better insight into the relationship between decimals and fractions. The place value chart below shows how the place value gets ten times smaller each time it shifts to the right. A whole unit divides into ten parts to make tenths, the tenths place divides further to give hundredths, and so on.

A place value chart from tens place to thousandths, with arrows showing division by ten when shifting to the right of each place.

Now let’s consider the decimal number 0.21 on the place value chart.

A place-value chart representing the decimal number 0.21

We have 2 tenths and 1 hundredth. If we add them up we get:

\frac{2}{10} + \frac{1}{100} =

\frac{2}{10} \times \frac{10}{10} + \frac{1}{100} \times \frac{1}{1} =

\frac{20}{100} + \frac{1}{100}=\frac{21}{100}

So the decimal 0.21 is

\frac{21}{100}

as a fraction. As the number of digits in the decimal number increases, the calculations get more tedious. Hence, the following step-by-step approach is more popular.

Let’s apply this to a few examples.

Example

Example 1. What is 0.375 as a fraction?

Answer:

Step 1. Consider the decimal as a fraction with a denominator of 1

0.375 = \frac{0.375}{1}

Step 2. Obtain an equivalent fraction with a whole number as the numerator

To convert 0.375 into a whole number, we need to shift the decimals 3 places to the left. Therefore, we multiply 0.375 by 10 three times, or simply by 1000.

\frac{0.375}{1} \times \frac{1000}{1000}=\frac{375}{1000}

Step 3. Write the fraction in its simplest form

The GCD of the numerator and denominator is 125. Divide the numerator and denominator by 125 to get:

\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}

Hence, 0.375 as a fraction is

\frac{3}{8}

Example

Example 2. Write 0.05 as a fraction.

Answer:

Step 1. Consider the decimal as a fraction with a denominator of 1

0.05 = \frac{0.05}{1}

Step 2. Obtain an equivalent fraction with a whole number as the numerator

Note here that the digit 5 is in the hundredths place and should shift two places to the left. So we should multiply it by 100.

\frac{0.05}{1} \times \frac{100}{100}=\frac{5}{100}

Step 3. Write the fraction in its simplest form

The GCD of the numerator and denominator is 5.

\frac{5 \div 5}{100 \div 5} = \frac{1}{20}

Hence, 0.05 as a fraction is

\frac{1}{20}

Example

Example 3. What is 1.5 as a fraction?

Answer:

Step 1. Consider the decimal as a fraction with a denominator of 1

1.5 = \frac{1.5}{1}

Step 2. Obtain an equivalent fraction with a whole number as the numerator

\frac{1.15}{1} \times \frac{10}{10}=\frac{15}{10}

Step 3. Write the fraction in its simplest form

The GCD of the numerator and denominator is 5.

\frac{15 \div 5}{10 \div 5} = \frac{3}{2} = 1\frac{1}{2}

Note the last equation follows from rules for the conversion of improper fractions to mixed fractions.

How to convert a fraction to decimal?

A fraction visually represents the division of the numerator by the denominator.

We can use the next example to understand how to turn a fraction into a decimal using division.

Example

Example 4. 3/8 as a decimal

Answer:

Step 1. Write the fraction in its simplest form This step is optional, yet it will make calculations easier to work out by hand.

In this example ⅜ cannot be simplified any further, so we can move on to step 2.

Step 2. Divide the numerator by the denominator. We have to divide 3 by 8. Let’s use long division to work it out.

Always ensure that the dividend is the denominator and the divisor is the numerator.

The first step of long division of 3 by 8 with 3 labeled as dividend and 8 as the divisor

Step 3. Obtain quotient after the remainder becomes 0

Continue the long division algorithm until there isn’t any remainder, as shown below.

Illustration of steps of dividing 3 by 8 using the long division method — 3 steps for dividing 3 by 8 using the long division method

Finally, we obtain that ⅜ is 0.375 as a decimal. Which tallies with the result we obtained in the very first example.

Note: The long division does not always yield a 0 remainder upon repeating, which happens with recurring decimals. In such cases, you can use the appropriate notation to represent the decimal.

It’s time to challenge ourselves to see how well we are able to convert a fraction into a decimal and vice versa.

Question

Review Question 1. Turn 0.625 into its decimal form

Solution:

Step 1. Consider the decimal as a fraction with a denominator of 1

0.625 = \frac{0.625}{1}

Step 2. Obtain an equivalent fraction with a whole number as the numerator

\frac{0.625}{1} \times \frac{1000}{1000}=\frac{625}{1000}

Step 3. Write the fraction in its simplest form

\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}

0.625 as a fraction is

\frac{5}{8}

Question

Review Question 2. Use a place value chart to write 3.5 as a fraction form

Solution:

Step 1. Write 3.5 on the place value chart

3 goes in the Unit's place, and 5 goes in the tenths place.

A place value chart representing the decimal number 3.5

Step 2. Add all the values as fractions

3 is a whole number.

3 + \frac{5}{10} =

\frac{3}{1} \times \frac{10}{10} + \frac{5}{10} \times \frac{1}{1}=

\frac{30}{10} + \frac{5}{10} = \frac{35}{10}

Step 3. Write the fraction in its simplest form

The GCD of 35 and 10 is 5. Hence we can simplify by dividing the numerator and denominator by 5.

\frac{35\div 5}{10 \div 5}= \frac{7}{2}= 3\frac{1}{2}

Hence, 3.5 as a fraction is

\frac{7}{2}

3\frac{1}{2}

as a mixed number.