this is a word problem:the country of hoffnemath is holding its presidential election this week. there three candidates(a,b,c,) and each eligible voter can vote for two of them. the candidate with the most total votes wins the election. here is a summary of the results:

>half of the eligible voters vote for candidate a,and divide their second vote among b and c in a ratio of 6:3.

> half of the other eligible voters cast their vote for candoidate b,and divide their second vote between a and cin the ratio 1:4

> two-thirds of the remaining eligible voters vote for c and divide their vote between a and b in the ratio 1:1.

>540 of the eligible voters do not vote at all.

1. how many votes does each candidate receive?

2. how many eligible voters are there in hoffmanmath country?

how do you use front end estimation to find the sum of 5.19 and 2.79

12+0 25+4.75

To solve this word problem, we need to break it down step by step and analyze the information given.

1. To find out how many votes each candidate receives, we need to calculate the number of votes each candidate receives from each group of eligible voters. Let's calculate it for each candidate:

a) Half of the eligible voters vote for candidate A, dividing their second vote among B and C in a ratio of 6:3.
- Let's assume the total number of eligible voters is 'x'.
- Half of the eligible voters, i.e., x/2, vote for candidate A.
- Out of those who vote for A, the ratio of their second vote between B and C is 6:3.
- Therefore, the number of votes A receives for the second vote is (x/2) * (6/9) = (x/2) * (2/3) = x/3.

b) Half of the other eligible voters cast their vote for candidate B. They divide their second vote between A and C in a ratio of 1:4.
- The remaining half of the eligible voters is also x/2.
- Out of those who vote for B, the ratio of their second vote between A and C is 1:4.
- Therefore, the number of votes B receives for the second vote is (x/2) * (1/5) = x/10.

c) Two-thirds of the remaining eligible voters vote for candidate C. They divide their vote between A and B in a ratio of 1:1.
- The remaining voters after the above two groups are (1 - 1/2 - 1/2) * x = (1/6) * x = x/6.
- Out of those who vote for C, the ratio of their vote between A and B is 1:1.
- Therefore, the number of votes C receives is (x/6) * (1/2) = x/12.

2. Now let's calculate the total votes for each candidate by adding up the votes from each group:
- Total votes for A = votes received for the first vote (x/2) + votes received for the second vote (x/3) = (x/2) + (x/3)
- Total votes for B = votes received for the first vote (x/2) + votes received for the second vote (x/10) = (x/2) + (x/10)
- Total votes for C = votes received for the first vote (x/6) + votes received for the second vote (x/12) = (x/6) + (x/12)

We need to simplify these expressions to get the final answers.

2. To find out the total number of eligible voters in Hoffmanmath country, we need to consider the votes received and the number of voters who did not vote.
- The total number of eligible voters = votes received by each candidate (A + B + C) + voters who did not vote.
- The problem states that 540 eligible voters did not vote at all. Hence, the total number of eligible voters is votes received + 540.

Let's calculate the answers using the equations we derived:

1. Votes for each candidate:
- Total votes for A = (x/2) + (x/3) = (3x + 2x) / 6 = 5x / 6
- Total votes for B = (x/2) + (x/10) = (5x + x) / 10 = 6x / 10 = 3x / 5
- Total votes for C = (x/6) + (x/12) = (2x + x) / 12 = 3x / 12 = x / 4

So, the number of votes each candidate receives is:
- Candidate A: 5x / 6
- Candidate B: 3x / 5
- Candidate C: x / 4

2. Total number of eligible voters:
- Total eligible voters = (votes for A + votes for B + votes for C) + voters who did not vote
- Total eligible voters = (5x / 6) + (3x / 5) + (x / 4) + 540

As we don't have any numerical values to compute the exact number of votes or eligible voters, we can only represent the answers in terms of 'x', the total number of eligible voters.