Today our topic of discussion is Similar and dissimilar repeating decimal fractions.
Similar and dissimilar repeating decimal fractions
Similar and dissimilar repeating decimal fractions
If for two or more repeating decimal fractions number of digits both in repeating and nonrepeating parts are the same, then they are called similar repeating decimal fractions. Otherwise they are called dissimilar repeating decimal fractions.
For example, 12.45 and 6.32; 9.453 and 125.897 are similar repeating decimal fractions. But, 0.3456 and 7.45789; 6.4357 and 2.89345 are dissimilar repeating decimal fractions.
Converting dissimilar repeating decimal fractions into similar ones
If we write repeating part of a repeating decimal fraction, the value does not change. For example, 6.4537 6.453737 = 6.453736.453737. Here each one of them is a repeating decimal fraction 6.45373737…, which is an infinite decimal number. If we convert this repeating decimal number into common fraction, we will see that all of them are equal.
In order to convert into similar repeating decimal fractions every repeating decimal must have equal number of digits in nonrepeating part. Repeating part must have number of digits equal to lowest common multiple of digits in all repeating parts.
Example 8. Convert 5.6, 7.345, and 10.78423 into similar repeating decimal fractions.
Solution: Repeating decimal-fractions 5.6, 7.345, and 10.78423 have nonrepeating digits equalling 0, 1 and 2 respectively. Here 10.78423 has the most number of digits in nonrepeating part, so in order to convert into similar repeating decimal-fractions we must have 2 digits in nonrepeating part of every number.
5.6, 7.345 and 10.78423 in repeating decimal-fractions have respectively 1, 2 and 3 digits in the repeating part. LCM of 1, 2 and 3 is 6. So in order to convert into similar repeating decimal-fractions we must have 6 digits in the repating part of every number.
So, 5.6 5.66666666, 7.345 7.34545454 and = 10.78423 10.78423423. = Desired similar repeating decimal-fractions are respectively 5.66666666, 7.34545454 and 10.78423423.
Example 9. Convert 1.7643, 3.24 and 2.78346 into similar repeating decimal-fractions.
Solution: In 1.7643 we have 4 digits in the nonrepeating part and it does not have repeating part. In 3.24 number of digits in nonrepeating part is 0 and that of repeating part is 2; in 2.78346 number of digits in nonrepeating part is 2 and that in repeating part is 3.
Here the largest number of nonrepeating digits is 4 and numbers of digits in the repeating parts are 2 and 3, LCM of which is 6. So each number must have 4 nonrep 2 ating digits and 6 repeating digits.
∴ 1.7643 1.7643000000, 3.243.2424242424 2.78346 = 2.7834634634 Desired similar repeating decimal-fractions are respectively 1.7643000000, 3.2424242424 and 2.7834634634.
Remarks: In order to convert finite decimal fractions into similar decimal fractions we must add sufficient number of Os at the end of the number to match with number of nonrepeating digits. In case of repeating decimal-fractions for each number, numbers of nonrepeating digits have been made equal so should be the case with repating digits.
Work: Convert 3.467, 2.01243 and 7.5256 into similar repeating decimal-fractions.
Read more:
