Question

When the repeating decimal 0.45454545…. is written in the simplest fraction form, the sum of the numerator and denominator is
(a) 5
(b) 11
(c) 55
(d) 16

Answer 1

Hint: Let the decimal number given to us be x. Multiply the equation you get in the previous step by 10 and 100 separately to get two more equations. Finally, subtract the two equations out of the three in which the RHS have the same decimal part to remove the decimals. Solve the equation you get to get the answer.

Complete step-by-step answer:
Let us start the solution to the above question by letting the decimal number given in the question to us be x. If we represent this relation in equation form, we get
 $ x=0.45454545\ldots .\text{ }...............\text{(i)} $
Now we will multiply this equation by 10. On doing so, we get
 $ 10x=10\times 0.45454545\ldots ... $
 $ 10x=4.54545454\ldots ....\text{ }..............\text{(ii)} $
Now we will again multiply the equation (i) by 100. On doing so, we get
 $ 100x=100\times 0.45454545\ldots ... $
 $ 100x=45.45454545\ldots ....\text{ }..............\text{(iii)} $
Now, if we see equation (i) and equation (iii), the RHS have the same decimal part, the RHS of equation one has 45 repeating in the decimal part one time more than equation (iii), but if we subtract 1 from infinity, we again get infinity.
So, we will subtract equation (i) from equation (iii). On doing so, we get
  $ 100x-x=\left( 45.45454545\ldots ....\text{ } \right)-\left( 0.454545..... \right) $
 $ \Rightarrow 99x=45 $
 $ \Rightarrow x=\dfrac{45}{99}=\dfrac{5}{11} $
So, the fractional form of 0.454545….. is $ \dfrac{5}{11} $ . Its denominator is 11 and numerator is 5. So, the sum of numerator and denominator is 5+11=16.
Hence, the answer to the above question is option (c).

Note: Remember that recurring decimal numbers are rational numbers and if you don’t get the decimal part same after the third equation as well, continue multiplying with higher powers of 10, i.e., 1000, 10000 and so on till you get the decimal part same as the RHS of equation (i) . The method is only valid for recurring decimal numbers.