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1.3 S) Converting Recurring Decimals to Fractions – Part 2
1.3 S) Converting Recurring Decimals to Fractions – Part 2
The content in this section builds on the content that was discussed in the previous section. Make sure that you have covered the content in the previous section before working through this section (click here to be taken back to the previous section).
Example 1
Convert the recurring decimal below into a fraction.
Convert the recurring decimal below into a fraction.
The first step is to let x equal this number and to write out a few more of the recurring figures. It is only the 3 for this number that recurs. Therefore, I am going to write this number out with a few more 3’s. This gives us the equation:
The next step is to multiply the equation by a multiple of 10. With the previous two questions, the recurring part of the numbers have started on the first decimal place. However, for this number, the recurring part of the number starts on the second decimal place. This means that we need to multiply the recurring number twice by two different multiples of 10. The aim of the first multiplication of a multiple of 10 is so that the recurring part of the number starts on the first decimal place. We are able to achieve this by multiplying x by 10. This results in us getting the equation below.
The aim of the second multiplication by a multiple of 10 is to get the recurring part of the multiple of 10 to overlap the recurring part of 10x; we want the overlapping part to be on the decimal places. We are able to achieve this when we have 100x. We can get 100x by multiplying x by 100 or by multiplying 10x by 10. The equations for x, 10x and 100x are shown below.
By looking at the above equations, we can see that the 3’s overlap one another in the decimal positions for 10x and 100x. Therefore, we can eliminate the recurring parts by taking 10x away from 100x. I am going to write 10x out below 100x and then complete the column subtraction. The working is shown below.
The value of x is our fraction and currently we have 90x. Therefore, we need to divide both sides of the equation by 90.
The final step is to check whether our fraction can be simplified. 21 and 90 both have a common factor of 3. Therefore, we divide the numerator and the denominator of the fraction by 3.
There are no more common factors between 7 and 30, which means that this fraction is in its simplest form.
Example 2
Convert the recurring decimal below into a fraction.
Convert the recurring decimal below into a fraction.
The first step is to let x equal this number and to write out a few more of the recurring digits. The two dots above the number indicates that we have a recurring group of figures and the recurring group of figures is the 6 and the 8. This gives us the equation:
We are now ready to multiply x by a multiple of 10. Like the previous question, the recurring part of this number starts on the second decimal place. Therefore, we need to multiply this number by a multiple of 10 so that the recurring part of the number starts on the first decimal place. We are able to achieve this by multiplying x by 10, which gives us the equation below:
We now need to multiply by a multiple of 10 so that the recurring parts of the numbers overlap one another in the decimal places. We are able to get the recurring parts to overlap in the decimal places by getting 1000x. We can get 1000x in two ways; one way is to multiply x by 1000 and the second way is to multiply 10x by 100. The equations for x, 10x and 1000x are shown below.
From looking at these 3 equations, we can see that the recurring parts overlap one another in the decimal places for 10x and 1000x. This means that we can eliminate the recurring parts by taking 10x away from 1000x. In order to make this calculation easier, I am going to write 10x below 1000x and column subtract. The working is shown below.
The value of x is our fraction and currently we have 990x. Therefore, we need to divide both sides of the equation by 990.
The final step is to check whether this fraction is in its simplest form, which we can do by looking for common factors between the numerator and the denominator. There is a common factor of 2 between 464 and 990, so we divide the numerator and the denominator of the fraction by 2.
There are no more common factors between 232 and 495, which means that the fraction is in its simplest form.