Recurring Decimals To Fractions
Neil Trivedi
Teacher
Contents
Recurring Decimals To Fractions
Terminating decimals are decimals that have a finite number of digits after the decimal point. Examples include or .
Recurring decimals are decimals in which one or more digits repeat indefinitely in a fixed pattern. We show the repeating pattern using dots above the digits that repeat (in some countries, they use a horizontal line above the digits).
For one recurring digit, we use a single dot. For example,
For two recurring digits, we indicate with a dot above each one. For example,
For three or more recurring digits, we indicate with a dot above the first and last digits that repeat. For example,
Recurring decimals can be written in exact form as fractions, and we will go through a method to do this conversion.
Converting Fractions to Recurring Decimals
To convert a fraction to a recurring decimal, we divide the numerator by the denominator using short division (also known as the bus stop method).
Example 1:
Rewrite the following fractions as recurring decimals.
a)
Single Step: Divide the numerator by the denominator using short division.
We start by dividing by . does not go into , so we write , and then put a decimal point. The remainder is , which we carry over, and we add a to make .
Next, we divide by . goes into two times, with a remainder of . We write in the decimal, carry over the remainder, and add another .
We now divide by again. Once again, goes into two times, with a remainder of . We write another in the decimal, carry over the remainder, and add another .
The same calculation keeps repeating, so the digit repeats in the decimal.
Therefore, the decimal is recurring and we get
b)
Single Step: Divide the numerator by the denominator using short division.
We start by dividing by . does not go into , so we write , and then put a decimal point. The remainder is , which we carry over, and we add a to make .
Next, we divide by . goes into six times, with a remainder of . We write in the decimal, carry over the remainder, and add another .
We now divide by . goes into three times, with a remainder of . We write in the decimal, carry over the remainder, and add another .
Now, we divide by again. This is the same calculation as before, so, we again get , with a remainder of . We write in the decimal, carry over the remainder and add another .
At this point, the remainders repeat then , so the digits and repeat in the decimal.
Therefore, the decimal is recurring and we get
Converting Recurring Decimals to Fractions
Recurring decimals can be written exactly as fractions. To do this, we use an algebraic method that removes the recurring digits by subtraction.
Converting a Recurring Decimal to a Fraction
1) Label the recurring decimal, calling it . Write out several digits to see the repeating pattern clearly.
2) Multiply by a power of i.e. , depending on the number of recurring digits.
If there is recurring digit, we multiply by .
If there is recurring digits, we multiply by .
If there is recurring digits, we multiply by , and so on.
3) Subtract the original equation from the new one. The recurring digits cancel out, leaving an equation with no recurring decimals.
4) Solve the equation for , writing the answer as a fraction and simplifying if possible.
Example 2:
Convert the following recurring decimals into fractions.
a)
Step 1: Label the recurring decimal.
We let
Step 2: Multiply by a power of , depending on the number of recurring digits.
Here, there is one recurring digit, so we multiply by .
Step 3: Subtract equation from equation .
Step 4: Solve for .
Dividing both sides by ,
Therefore,
b)
Step 1: Label the recurring decimal.
We let
Step 2: Multiply by a power of , depending on the number of recurring digits.
Here, there are two recurring digits, so we multiply by .
Step 3: Subtract equation from equation .
Step 4: Solve for .
Dividing both sides by ,
Therefore,
c)
Step 1: Label the recurring decimal.
We let
Step 2: Multiply by a power of , depending on the number of recurring digits.
Here, there are three recurring digits, so we multiply by .
Step 3: Subtract equation from equation .
Step 4: Solve for .
Dividing both sides by ,
This fraction can be simplified. In a non-calculator paper, we can quickly check if a number is divisible by and . We simply add the digits together and if that resulting number is divisible by and/or , then the original number is. With , if we add the digits , we get which is divisible by both and . This means that is divisible by both and . Therefore, to simplify quicker, we divide the numerator and denominator by .
Dividing the numerator and denominator by again,
Therefore,
We may be asked to perform the four operations (addition, subtraction, multiplication, and division) involving recurring decimals. In these cases, we first convert the recurring decimals into fractions and then carry out the operations using the fractions.
Example 3:
Use algebra to write the value of as a fraction in its simplest form.
We need to first convert both recurring decimals into fractions.
Starting with
Step 1: Label the recurring decimal.
We let
Step 2: Multiply by a power of , depending on the number of recurring digits.
Here, there are three recurring digits, so we multiply by .
Step 3: Subtract equation from equation .
Step 4: Solve for .
Dividing both sides by ,
Dividing the numerator and denominator by ,
Next with
Step 1: Label the recurring decimal.
We let
Step 2: Multiply by a power of , depending on the number of recurring digits.
Here, there are three recurring digits, so we multiply by .
Step 3: Subtract equation from equation .
Step 4: Solve for .
Dividing both sides by ,
Multiplying the numerator and denominator by to make the numerator a whole number,
Dividing the numerator and denominator by ,
Dividing the numerator and denominator by again,
Dividing the numerator and denominator by ,
Therefore, we have
Step 5: Add the fractions and simplify.
Rewrite both fractions so that they have a common denominator. In this case, the common denominator is the lowest common multiple of and , which is .
Adding the numerators,
Therefore,
Practice Question