Welcome to our Math lesson on Converting Recurrent Fractions into Decimals, this is the fourth lesson of our suite of math lessons covering the topic of Converting Fractions to Decimals and Vice-versa, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Converting Recurrent Fractions into Decimals
This is a very special part of this tutorial because at the first sight a recurrent decimal seems impossible to convert into a fraction due to its infinity of decimal places, so the conversion a recurring decimal into a fraction is impossible through the usual methods. Therefore, a completely new approach must be used for this kind of conversion. Let's begin to explain this new method through an example where a one-digit recurring decimal is involved. For example, let's consider the number 1.3333 (or 1.3); a recurring decimal obtained when dividing 4 by 3. The method we will explain here consists on the following steps:
Step 1: We denote the original number by N, i.e.
1.3333=N
Step 2: We multiply the original number by a factor that gives a new number with the same decimal part as the original. Here, this factor is 10. Thus, we have
13.3333 = 10 × N
Step 3: We subtract the original number from the new one to eliminate the recurring part, i.e.
13.3333-1.3333 = 12
The same thing is done with the factors containing N's. Thus, we have
10 × N - N = 9 × N
In this way, we obtain the equation
9 × N = 12
Step 4: Now, we solve the above equation and simplify the result is needed. Thus,
9 × N = 12
N = 12 ÷ 9
= 12/9
= (12 ÷ 3)/(9 ÷ 3)
= 4/3
This is the number we were referring to at the beginning of this explanation.
Let's consider a few other examples with other types of recurrence.
Example 6
Write the following recurring decimals as fractions.
a. 16.7777
b. 13.05050505
c. 5.324324324324
Solution 6
- We have
16.7777 = N
167.7777 = 10N
167.7777 - 16.7777 = 10N - N
151 = 9N
N = 151/9
- This time we have to multiply the original number by 100 to obtain again the same recurrence. Thus,
13.05050505 = N
1305.05050505 = 100N
1305.05050505 - 13.05050505 = 100N - N
1292 = 99N
N = 1292/99
- This time we have to multiply the original number by 1000 to obtain again the same recurrence. Thus,
5.324324324324 = N
5324.324324324324 = 1000N
5324.324324324324 - 5.324324324324 = 1000N - N
5319 = 999N
N = 5319/999
Both numerator and denominator are divisible by 9 (check this by using the divisibility rules explained in tutorial 1.6. Hence, we obtain N = (5319 ÷ 9)/(999 ÷ 9) = 591/111
Again, we see that both numbers are still divisible by 3, so N = (591 ÷ 3)/(111 ÷ 3) = 197/37
Sometimes the recurrence does not begin immediately after the decimal point but more on the right. In such cases, we first multiply the original number by a factor that leaves only the recurrence after the decimal point and then, we multiply this number by another suitable factor that gives the same recurrence, in order to make possible the elimination of recurrence through subtraction. Let's clarify this point though a couple of examples.
Example 7
Write the following recurring decimals as fractions.
a. 5.244444
b. 37.006666
c. 8.361616161
Solution 7
- We have
5.24 = N
52.4 = 10N
524.4 = 100N
524.4 - 52.4 = 100N - 10N
472 = 90N
N = 472/90
= (472 ÷ 2)/(90 ÷ 2)
= 236/45
- We have
37.006 = N
3700.6 = 100N
37006.6 = 1000N
37006.6 - 3700.6 = 1000N - 100N
33306 = 900N
N = 33306/900
= (33306 ÷ 3)/(900 ÷ 3)
= 11102/300
- We have
8.361 = N
83.61 = 10N
8361.61 = 1000N
8361.61 - 83.61 = 1000N - 10N
8278 = 990N
N = 8278/990
= (8278 ÷ 2)/(990 ÷ 2)
= 4139/495
A special case in this process is when the recurring part contains only 9s. In this case, we obtain not a fraction but a whole number as a result. Let's consider a couple of examples to clarify this point.
Example 8
Express the following recurring decimals as finite numbers.
a. 3.9
b. 219.9
Solution 8
- We have:
3.9 = N
39.9 = 10N
39.9 - 3.9 = 10N - N
36 = 9N
N = 36/9 = 4
- We have:
219.9 = N
2199.9 = 10N
2199.9 - 219.9 = 10N - N
1980 = 9N
N = 1980/9 = 220
More Converting Fractions to Decimals and Vice-versa Lessons and Learning Resources
Fractions Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 3.5 | Converting Fractions to Decimals and Vice-versa | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 3.5.1 | Decimal Fractions | | |
| 3.5.2 | The Meaning of Decimals. Converting Fractions into Decimals without a Calculator | | |
| 3.5.3 | Converting Decimals to Fractions | | |
| 3.5.4 | Converting Recurrent Fractions into Decimals | | |
Whats next?
Enjoy the "Converting Recurrent Fractions into Decimals" math lesson? People who liked the "Converting Fractions to Decimals and Vice-versa lesson found the following resources useful:
- Recurrent Fraction To Decimal Feedback. Helps other - Leave a rating for this recurrent fraction to decimal (see below)
- Fractions Math tutorial: Converting Fractions to Decimals and Vice-versa. Read the Converting Fractions to Decimals and Vice-versa math tutorial and build your math knowledge of Fractions
- Fractions Video tutorial: Converting Fractions to Decimals and Vice-versa. Watch or listen to the Converting Fractions to Decimals and Vice-versa video tutorial, a useful way to help you revise when travelling to and from school/college
- Fractions Revision Notes: Converting Fractions to Decimals and Vice-versa. Print the notes so you can revise the key points covered in the math tutorial for Converting Fractions to Decimals and Vice-versa
- Fractions Practice Questions: Converting Fractions to Decimals and Vice-versa. Test and improve your knowledge of Converting Fractions to Decimals and Vice-versa with example questins and answers
- Check your calculations for Fractions questions with our excellent Fractions calculators which contain full equations and calculations clearly displayed line by line. See the Fractions Calculators by iCalculator™ below.
- Continuing learning fractions - read our next math tutorial: The Meaning of Fractions. Equivalent Fractions. Simplifying Fractions
Help others Learning Math just like you
We hope you found this Math tutorial "Converting Fractions to Decimals and Vice-versa" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.
Fractions Calculators by iCalculator™