Question 1198174
<font color=black size=3>
Part (a)


I'll name the candidates A,B,C 
A = winner
B = runner up (aka 2nd place)
C = 3rd place candidate


The denominators of the fractions are 7 and 8. 
The LCD (lowest common denominator) is 7*8 = 56.


Consider the possibility there are 56 electors.
3/7 of 56 = (3/7)*56 = 24 electors voted for person A.


There are 56-24 = 32 electors who didn't vote for person A.
5/8 of which is (5/8)*32 = 20, and this is the number of people who voted for person B.


Therefore,
56-(24+20) = 56-44 = 12 people voted for person C.
12/56 = (3*4)/(14*4) = 3/14 of the electors voted for person C.


Another way to look at it:
3/7 voted for person A
1-3/7 = 4/7 didn't vote for person A
5/8 of 4/7 = (5/8)*(4/7) = 20/56 = 5/14 voted for person B
Then,
3/7 + 5/14 = 6/14+5/14 = 11/14 voted for either A or B
1 - 11/14 = 3/14 of the electors voted for C


-----------------------

Part (b)


A = amount winner got
B = amount 2nd place got
C = amount 3rd place got


"winner received 3021 votes more than the runner-up" means:
A = B+3021


3/7 voted for A.
4/7 didn't vote for A.
5/8 of 4/7 = 5/14 voted for B (calculated earlier)


x = total number of electors
A = 3/7 of x = (3/7)x = 3x/7 = 6x/14
B = 5/14 of x = (5/14)x = 5x/14


A = B+3021
6x/14 = 5x/14+3021
14 * [ 6x/14 ] = 14 * [ 5x/14+3021 ]
6x = 5x + 42294
6x-5x = 42294
x = 42294


-----------------------
<font color=red>Answers:</font>
(a) 3/14
(b) 42294
</font>