SOLUTION: In an election, there were three candidates. 3/7 of the electors voted for the winner. The runner-up received 5/8 of the remaining votes. a. What fraction of the electors voted fo

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: In an election, there were three candidates. 3/7 of the electors voted for the winner. The runner-up received 5/8 of the remaining votes. a. What fraction of the electors voted fo      Log On


   

Question 1198174: In an election, there were three candidates. 3/7 of the electors voted for the winner. The runner-up received 5/8 of the remaining votes.
a. What fraction of the electors voted for the third candidate?
b. If the winner received 3021 votes more than the runner-up, how many electors voted?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

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

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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

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Answers:
(a) 3/14
(b) 42294

Answer by ikleyn(52782) About Me  (Show Source):
You can put this solution on YOUR website!
.
In an election, there were three candidates. 3/7 of the electors voted for the winner.
The runner-up received 5/8 of the remaining votes.
a. What fraction of the electors voted for the third candidate?
b. If the winner received 3021 votes more than the runner-up, how many electors voted?
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                        Part   (a) 


Let x be the fraction under the problem's question (a).

Then you have this equation to find x

    3%2F7 + %285%2F8%29%2A%281-3%2F7%29 + x = 1,

    3%2F7 + %285%2F8%29%2A%284%2F7%29 + x = 1,

    3%2F7 + 5%2F%282%2A7%29 + x = 1,

    3%2F7 + 5%2F14 + x = 1,

    6%2F14 + 5%2F14 + x = 1,

    11%2F14 + x = 1,

    x = 1 - 11%2F14 = 3%2F14.    ANSWER to question (a)

Part  (a)  is solved.


                        Part   (b) 


Let x be the number of voters.


We are given

    %283%2F7%29x - %285%2F8%29%2A%284%2F7%29%2Ax = 3021.


Simplify and find x.  First step multiply all the terms by 7*8 = 56

    24x - 5*4*x = 56*3021

    24x - 20x   = 56*3021

        4x      = 56*3021

         x    =   %2856%2F4%29%2A3021 = 14*3021 = 42294  voters.       ANSWER to question (b)

Solved.