in an election, there were three candidate. 3/7 of the voters voted for the winner. The runner up received 5/8 of the remaining votes. (a)What fraction of the voters voted for the third candidate? (b)If the winner received 3021 votes more than the runner up, how many voters voted?
4/7 did not vote for the winner
... 5/8 of these voted for the runner up
... runner up's portion ... 4/7 * 5/8 = ?
3rd candidate ... 1 - winner - runner up = 1 - 3/7 - runner up = ?
v (3/7 - runner up) = 3021 ... v = ?
To determine the fractions of voters who voted for the third candidate and the total number of voters, we need to break down the given information step by step.
Let's assign variables to the unknown quantities:
- Let W be the fraction of voters who voted for the winner.
- Let R be the fraction of voters who voted for the runner up.
- Let T be the fraction of voters who voted for the third candidate.
We also know that:
1) W + R + T = 1 (since all voters must support one of the three candidates).
Given that 3/7 of the voters voted for the winner, we can write:
W = 3/7
After the winner is determined, the remaining votes are distributed between the runner-up and the third candidate. We are told that the runner up received 5/8 of the remaining votes. Since the remaining votes are represented by (1 - W) (since W is the fraction of votes for the winner), we can write:
R = 5/8 * (1 - W)
Now, let's solve for T using the information above.
Since W + R + T = 1, we can substitute the values of W and R into the equation:
3/7 + 5/8 * (1 - 3/7) + T = 1
To solve the equation and find the value of T, we need to simplify it:
3/7 + 5/8 * 4/7 + T = 1
3/7 + 20/56 + T = 1
24/56 + 20/56 + T = 1
Combining the fractions:
44/56 + T = 1
To isolate T, subtract 44/56 from both sides:
T = 1 - 44/56
T = 56/56 - 44/56
T = 12/56
T = 3/14
Therefore, (a) 3/14 of the voters voted for the third candidate.
Now, let's move on to part (b):
We are told that the winner received 3021 votes more than the runner up. Let's assume the number of votes for the runner up is represented by x.
Therefore, the number of votes for the winner would be x + 3021.
To find the total number of voters, we need to determine the value of x (votes for the runner up).
We know that:
W + R + T = 1
Substituting the values of W and R:
3/7 + 5/8 * (1 - 3/7) + 3/14 = 1
To simplify and solve:
24/56 + 20/56 + 12/56 = 1
56/56 = 1
This equation confirms that our previous calculations are correct.
Next, we'll convert the fractions to a common denominator of 56 and combine them:
(24 + 20 + 12)/56 = 1
56/56 = 1
Therefore, the total number of voters is the same as the common denominator, 56.
Now, let's find the value of x (votes for the runner up):
x + 3021 = votes for the winner
x = votes for the runner up
Since the winner received 3/7 of the votes and the remaining votes are split between the runner up and the third candidate, we can write the equation:
3/7 = x + 3021 + 3/14
To solve for x, we need to simplify the equation:
6/14 = x + 3021 + 3/14
6/14 = x + 3024/14
Combine the fractions:
6/14 = (x + 3024)/14
Since the denominators are the same, we can equate the numerators:
6 = x + 3024
To isolate x, subtract 3024 from both sides:
6 - 3024 = x
-3018 = x
Therefore, the runner up received -3018 votes. However, negative votes are not possible, so there must be an error in the calculations or the problem statement. Please check the given information and calculations to find the correct solution.