Today our topic of discussion is Addition and subtraction of repeating decimal fractions.
Addition and subtraction of repeating decimal fractions
Addition and subtraction of repeating decimal fractions In order to add or subtract repeating decimal fractions, we need to convert repeating decimal fractions into similar repeating decimals. Then addition or subtraction must be carried out as in case of finite decimal fractions.
If addition or subtraction of finite and repeating decimal fractions together are done, in order to make repeating decimal factions similar, the number of digits of non-repeating part of each repeating fraction should be equal to the number of digits between the numbers of digits after the decimal points of finite decimal fractions and that of the non-repeating parts of repeating decimal fraction.
The number of digits of repeating part will be equal to L.C.M. as obtained by applying the rules and in case of finite decimal fractions, necessary numbers of zeros are to be used in its repeating parts. Then the same process of addition and subtraction is to be done following the rules of finite decimal fractions. The sum or the difference thus obtained will not be the actual one.
It should be observed that in the process of addition of similar decimal fractions if any number is to be carried over after adding the digits at the extreme left of the repeating part of the decimal fractions, then that number is added to the sum obtained and thus the actual sum is found.
In case of subtraction the number to be carried over is to subtract from the difference obtained and thus actual result is found. The sum or difference thus found is the required sum or difference.
Remarks:
1. The sum or difference of repeating decimal is also repeating decimal fractions. In this sum or difference the number of digits in the non-repeating part will be equal to the number of digits in the non- repeating part of that repeating decimal fractions, which have the highest number of digits in its non-repeating part.
Similarly, the number of digits in the repeating part of the sum or the result of subtraction will be the equal to L.C.M. of the numbers of digits of repeating parts of repeating decimal fractions. If there are finite decimal fractions, the number of digits in the nonrepeating part of each repeating decimal fraction will be equal to the highest numbers of digits that occurs after the decimal point.
2. Converting the repeating decimal fractions into common fractions, addition and subtraction may be done according to the rules as used in case of fractions and the sum or difference is converted into decimal fractions. But this method of addition or subtraction needs more time.
Example 10. Add 3.89, 2.178 and 5.89798.
Solution: We have 2 digits in nonrepeating part, and repeating parts have respectively 2, 2 and 3 for which LCM is 6. At first all these three reepating decimal fractions have been converted into similar repeating decimal fractions.
3.89 = 3.89898989
2.178 =2.17878787
5.89798 =5.89798798
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11.97576574 [ 8 + 8 + 7 + 2 = 25 here 2 is 2 in hand here
+ 2 2 of 25 has been added]
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11.97576576
The desired result is 11.97576576 or, 11.97576
Remarks: In the result 576576 is repeating part. But if we assume 576 in the repeating part, then there will be no change of value.
Note: In order to clarify addition at the rightmost digit we have solved the addition in a different way:
3.89 = 3.89898989|89
2.178 = 2.17878787|87
5.89798 = 5.89798798|79
_______________________________________
11.97576576|55
Here after the end of repe ating part some more digits have been taken. These extra digits have been separated by a vertical line. Then the addition has been performed. From the addition of digits just after the vertical line 2 in hands has been added to the digits just before the vertical line. It may be noted here the digit on the right side of the vertical line is the same as the first digit of repeatation. Therefore both the sums ore the same.
Example 11. Add 8.9478, 2.346 and 4.71.
Solution: In order to make repeating decimal fractions similar we must have 3 nonrepeating digits and repeating part must have digits equal to LCM of 2 and 3, that is 6. Now decimal fractions will be added.
8.9478 =8.947847847
2.346 =2.346000000
4.71 =4.717171717
______________________________________
16.011019564 [ 8 + 0 + 1 + 1 = 10 Here 1 is 1 in hand
+1 Here 1 of 10 has been added]
__________________________________________
16.011019565
The desired sum is 16.011019565
Work: Add 1): 2.097 and 5.12768 2) 1.345, 0.31576 and 8.05678
Example 12. Subtract 5.24673 from 8.243.
Solution: Here number of digits in nonrepeating-part should be 2 and that in repeating part should be 6, LCM of 2 and 3. Now the numbers have been converted into similar numbers, and subtraction has been performed as follows:
8.243 = 8.24343434
5.24673 = 5.24673673
___________________________
2.99669761 [subtracting 6 from 3 results in 1 in hand]
– 1
___________________________
2.99669760
The desired result is 2.99669760.
Remarks: If the minuend starting at the repeating dot is smaller than the subtrahend, we must subtract 1 from the rightmost digit.
Example 13. Subtract 16.437 from 24.45645.
Solution:
24.45645 = 24.45645
16.437 = 16.43743
________________________
8.01902 [if 7 is subtracted from 6, we have 1 in hand]
– 1
_________________________
8.01901
The desired result is 8.01901.
Note: In the following is explained why 1 is subtracted from the rightmost digit in different way.
24.45645 =24.45645|64
16.437 =16.43743|74
_________________________
8.01901|90
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