Question

If $\dfrac{1}{{891}} = 0.00112233445566778899$………., then what is the value of $\dfrac{{198}}{{891}}$?
A. $0.222$……
B. $0.111$……
C. $0.3232$…..
D. $0.4242$….

Answer 1

Hint: In the given question, you have given a fraction and the value of the fraction in decimals. We have to find the value of another fraction in decimals whose denominator is the same as the previous one. We can calculate it by comparing it with the first one i.e. another fraction is 198 times the first fraction whose decimal conversion is given.
\[ \Rightarrow \dfrac{{198}}{{891}} = 198 \times \dfrac{1}{{891}}\]
\[ \Rightarrow \dfrac{{198}}{{891}} = 198 \times 0.00112233445566778899\]…………..

Complete step by step answer:

Let us see what is given to us? We have given a fraction and its decimal conversion i.e.
$ \Rightarrow \dfrac{1}{{891}} = 0.00112233445566778899$…… (1)
After that see what we have to find? We have to find the decimal value of another fraction i.e. $\dfrac{{198}}{{891}}$
Split this fraction in integer number and fraction as given below
\[ \Rightarrow \dfrac{{198}}{{891}} = 198 \times \dfrac{1}{{891}}\]…… (2)
Comparing equation (1) and (2), we get
\[ \Rightarrow \dfrac{{198}}{{891}} = 198 \times 0.00112233445566778899\]……
As it takes time to multiply these two numbers and chances of errors are there. So, we can also solve it by splitting the decimal number as follows:
$ \Rightarrow 0.00112233445566778899$… $ = 0.0011 + 0.000022 + 0.00000033$……..
$ \Rightarrow 198 \times 0.00112233445566778899$…$ = 198 \times (0.0011 + 0.000022 + 0.00000033$…)
Now to find $198 \times 0.0011$, we can do multiplication of $198 \times 11$. Here to multiply with 11 we can use a trick given as follow
 

Therefore, $198 \times 0.0011 = 0.2178$
$ \Rightarrow 198 \times 0.000022 = 0.004356$
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So on.

$ \Rightarrow 198 \times 0.00112233445566778899$…... =$198 \times 0.0011 + 198 \times 0.000022 + $…..
      $ = 0.2178 + 0.004356 + $………..
$ \Rightarrow 198 \times 0.00112233445566778899$……..$ = 0.222$

So, the correct option is A.

Note: In these types of questions generally students take approximate value, this is also the right way but if options are given to us. If options are not given to us or the two digits after the decimal are the same then don’t use approximation.
Sometimes students leave this question in multiple-choice questions, thinking it may be time-consuming, and while calculating this whole the chances of mistakes are more. But you use the given trick then you can avoid mistakes that occur during calculations.