Question

In a city $40\% $ of the adults are illiterate while $85\% $ of the children are literate. If the ratio of the adults to that of the children is $2:3$ then what percent of the population is literate?
A. $20\% $
B. $25\% $
C. $50\% $
D. $60\% $
E. $75\% $

Answer 1

Hint: Let us assume a number to convert the ratio into the number. Further, we will convert adults illiterate to literate. Thereafter, we will find a literate population, then we will calculate the percentage of the population that is literate.


Complete step by step solution:
 The given ratio between the adults and the children $ = 2:3.$
Let the number of adults be $ = 2x$
And number of children be $ = 3x$
Then, $40\% $ of the adults are illiterate and $85\% $ of the children are literate.
Now, we will convert illiterate $40\% $ adults into literate then $\left( {100\% - 40\% } \right)$
 $40\% $ adults into literate then $ = 60\% $
According to the given information in the question.
Literate population $ = 60\% $ of adult’s literate $ + 85\% $ of children literate.
Literate population $ = 60\% \,\,of\,2x + 85\% \,\,of\,3x$
Literate population $ = 60\% \,\, \times \,2x + 85\% \,\, \times \,3x$
Literate population $ = \dfrac{{60}}{{100}}\,\, \times \,2x + \dfrac{{85}}{{100}}\,\, \times \,3x$
Literate population $ = \dfrac{6}{{10}}\,\, \times \,2x + \dfrac{{17}}{{20}}\,\, \times \,3x$
Literate population $ = \dfrac{{6 \times \,2x}}{{10}}\,\, + \dfrac{{17 \times \,3x}}{{20}}\,\,$
Literate population $ = \dfrac{{12x}}{{10}}\,\, + \dfrac{{51x}}{{20}}\,\,$
We will take LCM of $10,20 = 20,$we have
Literate population $ = \dfrac{{2 \times 12x + 51x}}{{20}}$
Literate population $ = \dfrac{{24x + 51x}}{{20}}$
Literate population $ = \dfrac{{75x}}{{20}}$
Now, required percentage \[ = \left[ {\left( {\dfrac{{75x}}{{20}}} \right) \times \left( {\dfrac{1}{{2x + 3x}}} \right) \times 100} \right]\% \]
Required percentage \[ = \left[ {\left( {\dfrac{{75x}}{{20}}} \right) \times \dfrac{1}{{5x}} \times 100} \right]\% \]
Required percentage\[ = \left[ {\left( {\dfrac{{75}}{{20}}} \right) \times \dfrac{1}{5} \times 100} \right]\% \]
Required percentage\[ = \left[ {\left( {\dfrac{{75}}{2}} \right) \times \dfrac{1}{5} \times 10} \right]\% \]
Required percentage\[ = \left[ {\left( {75} \right) \times \dfrac{1}{5} \times 5} \right]\% \]
Required percentage\[ = \left[ {75} \right]\% \]
Thus, the required percentage is \[75\% \].
Hence, the correct option is E.


Note: Students must know that while you will find the literate population then you should convert adults literate and children literate.