In an election a voter can vote for 2 candidates at a time. Half of the voters give their `1^(st)` vote to candidate A and the 2nd vote to candidate B, C and D in ratio 3:2 : 1. Half of the rest voters give their `1^(st)` vote to B and their `2^(nd)` vote to C and D in ratio 2:1. Half of the rest of the voters give their `1^(st)` vote to C and D in ratio 1:1.840 voters do not cast their vote. Find the number of votes each candidate got.

Options

A. `3360, 3360, 308, 19 60`
B. `3000, 3100, 3080, 1960`
C. `3360, 3200, 1100, 1800`
D. `3160, 3000, 1000, 1100`

Answer

To solve the problem, we will break down the voting process step by step.

### Step 1: Determine the total number of voters
Let the total number of voters be \( V \). According to the problem, 840 voters did not cast their votes. Therefore, the number of voters who did cast their votes is:
\[ V - 840 \]

### Step 2: Votes for Candidate A
Half of the voters gave their first vote to candidate A. Therefore, the number of voters who voted for A is:
\[ \frac{1}{2}(V - 840) \]

### Step 3: Distribution of second votes for B, C, and D
The second votes of these voters are distributed among candidates B, C, and D in the ratio 3:2:1. The total parts in this ratio are:
\[ 3 + 2 + 1 = 6 \]
Thus, the second votes for B, C, and D can be calculated as follows:
- Votes for B:
\[ \frac{3}{6} \times \frac{1}{2}(V - 840) = \frac{1}{4}(V - 840) \]
- Votes for C:
\[ \frac{2}{6} \times \frac{1}{2}(V - 840) = \frac{1}{6}(V - 840) \]
- Votes for D:
\[ \frac{1}{6} \times \frac{1}{2}(V - 840) = \frac{1}{12}(V - 840) \]

### Step 4: Remaining voters
After the first half of the voters, the remaining voters are:
\[ \frac{1}{2}(V - 840) \]

### Step 5: Votes for B, C, and D from the remaining voters
Half of the remaining voters (which is \( \frac{1}{4}(V - 840) \)) give their first vote to B and distribute their second votes to C and D in the ratio 2:1. The total parts in this ratio are:
\[ 2 + 1 = 3 \]
Thus, the votes for B, C, and D from these voters are:
- Votes for B:
\[ \frac{2}{3} \times \frac{1}{4}(V - 840) = \frac{1}{6}(V - 840) \]
- Votes for C:
\[ \frac{1}{3} \times \frac{1}{4}(V - 840) = \frac{1}{12}(V - 840) \]

### Step 6: Remaining voters again
The remaining voters after this round are:
\[ \frac{1}{4}(V - 840) \]

### Step 7: Votes for C and D from the last remaining voters
Half of these remaining voters (which is \( \frac{1}{8}(V - 840) \)) give their first vote to C and distribute their second votes to D in the ratio 1:1. Thus, the votes for C and D from these voters are:
- Votes for C:
\[ \frac{1}{2} \times \frac{1}{8}(V - 840) = \frac{1}{16}(V - 840) \]
- Votes for D:
\[ \frac{1}{2} \times \frac{1}{8}(V - 840) = \frac{1}{16}(V - 840) \]

### Step 8: Total votes for each candidate
Now, we can sum up the votes for each candidate:
- Total votes for A:
\[ \frac{1}{2}(V - 840) \]
- Total votes for B:
\[ \frac{1}{4}(V - 840) + \frac{1}{6}(V - 840) = \left(\frac{3}{12} + \frac{2}{12}\right)(V - 840) = \frac{5}{12}(V - 840) \]
- Total votes for C:
\[ \frac{1}{6}(V - 840) + \frac{1}{12}(V - 840) + \frac{1}{16}(V - 840) = \left(\frac{2}{12} + \frac{1}{12} + \frac{3/48}\right)(V - 840) \]
- Total votes for D:
\[ \frac{1}{12}(V - 840) + \frac{1}{16}(V - 840) = \left(\frac{4}{48} + \frac{3/48}\right)(V - 840) \]

### Step 9: Solve for V
Now we know that the total number of voters who cast votes is:
\[ \frac{1}{2}(V - 840) + \frac{5}{12}(V - 840) + \frac{1}{6}(V - 840) + \frac{1}{16}(V - 840) + \frac{1}{16}(V - 840) \]
This equals:
\[ \frac{240}{240}(V - 840) = V - 840 \]
Now, we can set up the equation:
\[ \frac{1}{2}(V - 840) + \frac{5}{12}(V - 840) + \frac{1}{6}(V - 840) + \frac{1}{16}(V - 840) + \frac{1}{16}(V - 840) = V - 840 \]

### Step 10: Final calculation
After solving for \( V \), we can find the total votes for each candidate.