In this explainer, we will learn how to convert a recurring decimal to a fraction or a mixed number.
A fraction, in everyday language, often means a tiny part. In math, a fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction is the number of equal shares, or “portions,” the whole is split into, while the numerator is the number of these shares that make the part we are considering.
But a fraction can also be understood as a quotient: it is simply dividing one quantity by another. Any rational number can be expressed in this way, and in this case a “fraction” is actually a number.
Let us take as an example.
Fraction: If is a fraction, it means that we are comparing a part made of 3 equal shares to the whole made of 4 such equal shares.
If we consider the whole to be the unit (1), then the part is 0.75.
Number: If is a number obtained by dividing 3 by 4, then it is 0.75.
We see that the rational number (3 divided by 4) is the same as (three-quarters) of 1. They can be both expressed as a decimal, 0.75.
It is important to remember the definition of a rational number.
Definition: Rational Number
A rational number is a real number that can be expressed as a simple fraction (i.e., whose denominator and numerator are integers).
As we have seen above with , a rational number can be expressed as a decimal. We are going to learn here how to convert a decimal back to a fraction (of 1). For instance, consider 0.7. It is seven-tenths (i.e., ). 0.03 is three-hundredths (i.e., ). And note that 2 is simply .
We can see here that our decimal number system makes it very easy to convert a decimal back into a fraction with a power of ten as the denominator.
Consider, for instance, 1.372. This decimal can be split as
- 1 unit,
- 3 tenths,
- 7 hundredths,
- 2 thousandths.
The smallest bit in this partition of 1.372 is the two-thousandths. We will, therefore, express 1.372 as a number of thousandths:
- 1 unit is one thousand-thousandths, written as .
- 3 tenths are three hundred-thousandths, written as .
- 7 hundredths are seventy-thousandths, written as .
- 2 thousandths are written as .
Adding all the parts together, we get
We see that, in this method, the position of the last digit of a number determines the denominator of the fraction. A last digit in the third position after the decimal point means that it is in the thousandths column, so the simplest denominator to start with is 1,000. Then, to find the denominator, we can simply multiply our number by the value of the denominator:
So far, we have converted a decimal to a fraction. Now, we may want to express this fraction in its simplest form. We can do this in our example by simplifying by two twice: . This fraction cannot be simplified any further since 343 is a multiple of neither 5 nor 2, which are the only two prime factors of 250.
This fraction is a so-called improper fraction since its numerator is greater than its denominator. It can, therefore, be written as a mixed number:
We are going to look at some examples to check our understanding.
Example 1: Converting a Simple Decimal to a Fraction
Convert 0.4 to a fraction.
In the decimal 0.4, the last digit, 4, is in the tenths column, meaning that 0.4 is 4 tenths. Hence, .
Now that we have converted 0.4 to a fraction, we need to write the fraction in its simplest form. As both 4 and 10 are multiples of 2, the fraction can be simplified by dividing by 2: .
Therefore, our answer is .
The next example is slightly more complex.
Example 2: Converting a Decimal with Three Decimal Places to a Fraction
Convert 0.268 to a fraction.
In the decimal 0.268, the last digit is 8 and it is at the third decimal place. This means that it is in the thousandths column. Therefore, 0.268 can be expressed as a whole number of thousandths: .
Now that we have converted 0.268 to a fraction, we need to write the fraction in its simplest form. The numbers 268 and 1,000 are both multiples of 4; therefore, by dividing both the numerator and denominator by 4, we find .
Our answer is .
In the next example, we are going to look at how to express a decimal as a mixed number.
Example 3: Converting a Decimal to a Mixed Number
The age of Earth is about 4.54 billion years. Write this as a mixed number in its simplest form.
The age of Earth is given here as a number of billion years, and it is given as a decimal, 4.54. We are asked to write 4.54 as a mixed number in its simplest form.
First, we convert 4.54 into a fraction: .
Then, we express this improper fraction as a mixed number: .
Finally, we find the simplest form of the fractional part. As both 54 and 100 are multiples of 2, we can divide both the numerator and denominator by 2: . There is no common factor between 27 and 50 except 1; therefore, this fraction is in its simplest form.
Hence, the answer is .
So far, we have looked at examples with terminating decimals. Now, we are going to learn how to convert repeating decimals to a fraction. A repeating decimal occurs when the division never ends. The bar notation is often used to indicate that a digit or group of digits is repeated. For instance, is written as , and is written as .
We know that , but how can we find, having only , that it can be written as ? We are going to learn a simple method based on the realization that subtracting two repeating decimals that have exactly the same decimal part repeating after the decimal point leads to an integer.
Let us start with the simple case of . For the sake of clarity, we are going to call this number . If we multiply by 10, we find another number with exactly the same repeating decimal: . And if we subtract from , we find which can be simplified to
From this, by dividing both sides of the equation by 9, the value of can be expressed as
Let us look at two more complex examples to deepen our understanding of the method.
Example 4: Converting a Repeating Decimal to a Fraction
Express as a rational number in its simplest form.
Let be . To express as a fraction in its simplest form, we need first to identify the repeating part: it is the digits with the bar on the top, so it is the group of digits “75.” We have to multiply by a power of 10 so that the result is a number with the same repeating decimals (i.e., “75”). We see that , and so which can be simplified to From this, it follows that which can be simplified by dividing both the numerator and denominator by 3 to
In the next example, the repeating decimals do not start just after the decimal point.
Example 5: Converting a Repeating Decimal to a Fraction
Convert to a fraction.
Let be the number . To express as a fraction in its simplest form, we need first to identify the repeating part: it is the digit with the bar on the top, “7.” And we notice that the repeating decimal does not start straight after the decimal point. In this case, we will have to multiply by two different powers of 10, so that both results are a number with the same repeating decimal starting straight after the decimal point.
The smallest power of 10 that will work is 100:
For the sake of simplicity, we can choose to multiply by the next power of 10, which is 1,000. We get
We can now subtract the smaller from the larger: which can be simplified to
From this, it follows that
This fraction cannot be simplified; it is therefore the final answer.
- A fraction can be understood as a quotient: it is simply dividing one quantity by another. Any rational number can be expressed in this way, and in this case a “fraction” is actually a number. If is a number obtained by dividing 3 by 4, then it is 0.75.
- We see that the rational number (3 divided by 4) is the same as (three-quarters) of 1. They can both be expressed as a decimal, 0.75.
- Converting a decimal to a fraction is finding the quotient that gives this decimal. To do this, we use the properties of our decimal system. A terminating decimal can always be expressed as a whole number divided by a power of 10. For instance, .
- To convert a repeating decimal, , to a fraction, the method
consists of multiplying by two different powers of 10 so that we get
two decimals with exactly the same repeating decimal part. They can be subtracted from
each other to give a whole number. This allows us to find the quotient that gives
, as illustrated in the following two examples.
- First example: . We take and then subtract from : which simplifies to and gives (by dividing both sides by 99) Note that, in this example, we could use and did not need to multiply it by a power of 10.
- Second example: . We take and , and subtract from : which simplifies to and gives (by dividing both sides by 900)