Question

\[\dfrac{2}{5}\] of the voters promise to vote for P and the rest promised to vote for Q. Of these on the last day 15 % of the voters went back of their promise to vote for P and 25 % of voters went back of their promise to vote for Q, and P lost by 2 votes. Then, what is the total number of voters?
(a). 100
(b). 110
(c). 90
(d). 95

Answer 1

Hint: Assume a variable x for the total number of voters. Find the number of votes got by P, out of \[\dfrac{2}{5}\] of the voters 15% of voters went back of their promise to vote for P means they didn't voted P and 25% of voters went back of their promise to vote for Q means they voted P then find final votes got by P and Q, their difference of votes is given as 2. So,write down the equations and find the total number of voters by solving the equation.

Complete step-by-step answer:
Let the total number of voters be x. It is given that \[\dfrac{2}{5}\] of the voters promise to vote for P and the rest for Q. Hence, we have:
\[P = \dfrac{2}{5}x..............(1)\]
\[Q = x - \dfrac{2}{5}x\]
\[Q = \dfrac{3}{5}x.............(2)\]
It is given that on the last day, 15 % of the voters went back of their promise to vote for P. Hence, we have:
\[P' = P - (15\% ofP)\]
Using equation (1), we have:
\[P' = \dfrac{2}{5}x - \left( {\dfrac{{15}}{{100}} \times \dfrac{2}{5}x} \right)\]
Simplifying, we have:
\[P' = 0.4x - \left( {0.06x} \right)\]
\[P' = 0.34x..........(3)\]
\[Q' = Q + (15\% ofP)\]
Using equation (1) and equation (2), we have:
\[Q' = \dfrac{3}{5}x + \left( {\dfrac{{15}}{{100}} \times \dfrac{2}{5}x} \right)\]
\[Q' = 0.6x + 0.06x\]
\[Q' = 0.66x............(4)\]
It is also given that 25 % of the voters went back of their promise to vote for Q. Hence, we have:
\[P'' = P' + (25\% ofQ)\]
Using equation (2) and equation (3), we have:
\[P'' = 0.34x + \left( {\dfrac{{25}}{{100}} \times \dfrac{3}{5}x} \right)\]
Simplifying, we have:
\[P'' = 0.34x + 0.15x\]
\[P' = 0.49x..........(5)\]
\[Q'' = Q' - (25\% ofQ)\]
Using equation (2) and equation (4), we have:
\[Q'' = 0.66x - \left( {\dfrac{{25}}{{100}} \times \dfrac{3}{5}x} \right)\]
\[Q'' = 0.66x - 0.15x\]
\[Q'' = 0.51x............(6)\]
It is given that P lost by 2 votes. Hence, we have:
\[Q'' - P'' = 2\]
Using equation (5) and equation (6), we have:
\[0.51x - 0.49x = 2\]
Simplifying, we have:
\[0.02x = 2\]
Solving for x, we have:
\[x = \dfrac{2}{{0.02}}\]
\[x = 100\]
Hence, the total number of voters is 100.
Hence, the correct answer is option (a).

Note: You can also just find the final number of voters for P and then subtract it from the total number of voters to find the final number of voters for Q and then proceed with the solution.