There are 3 contestants P, Q and R an election. P secured 20% of the votes and Q secured 70% of the remaining voltes. If R secured 4800 votes, by how many votes has the winner won the election?

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the total number of votes
Let the total number of votes be \( V \).

### Step 2: Calculate the votes secured by P
P secured 20% of the total votes. Therefore, the votes secured by P can be calculated as:
\[ \text{Votes by P} = 0.20 \times V = \frac{V}{5} \]

### Step 3: Calculate the remaining votes after P's share
The remaining votes after P's share can be calculated as:
\[ \text{Remaining Votes} = V - \text{Votes by P} = V - \frac{V}{5} = \frac{4V}{5} \]

### Step 4: Calculate the votes secured by Q
Q secured 70% of the remaining votes. Therefore, the votes secured by Q can be calculated as:
\[ \text{Votes by Q} = 0.70 \times \text{Remaining Votes} = 0.70 \times \frac{4V}{5} = \frac{28V}{50} = \frac{14V}{25} \]

### Step 5: Calculate the votes secured by R
We know that R secured 4800 votes. Therefore, we can express R's votes as:
\[ \text{Votes by R} = 4800 \]

### Step 6: Set up the equation for total votes
The total votes can be expressed as the sum of votes secured by P, Q, and R:
\[ \text{Votes by P} + \text{Votes by Q} + \text{Votes by R} = V \]
Substituting the values we calculated:
\[ \frac{V}{5} + \frac{14V}{25} + 4800 = V \]

### Step 7: Solve for V
To solve for \( V \), we need a common denominator. The common denominator for 5 and 25 is 25. Rewriting the equation:
\[ \frac{5V}{25} + \frac{14V}{25} + 4800 = V \]
Combining the fractions:
\[ \frac{19V}{25} + 4800 = V \]
Now, isolate \( V \):
\[ 4800 = V - \frac{19V}{25} \]
\[ 4800 = \frac{25V - 19V}{25} \]
\[ 4800 = \frac{6V}{25} \]
Multiplying both sides by 25:
\[ 120000 = 6V \]
Dividing by 6:
\[ V = 20000 \]

### Step 8: Calculate votes for P and Q
Now that we have \( V \), we can calculate the votes for P and Q:
- Votes by P:
\[ \text{Votes by P} = \frac{V}{5} = \frac{20000}{5} = 4000 \]
- Votes by Q:
\[ \text{Votes by Q} = \frac{14V}{25} = \frac{14 \times 20000}{25} = 11200 \]

### Step 9: Determine the winner and the margin
The winner is Q with 11200 votes. Now, we can find out by how many votes the winner won:
\[ \text{Margin} = \text{Votes by Q} - \text{Votes by R} = 11200 - 4800 = 6400 \]

### Final Answer
The winner has won the election by **6400 votes**.

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