Chamanlal, Arshad and Jagit Singh contested an election. All the votes polled were valid. Arshad got 35% of the total votes. For every 35 votes Chamanlal got 14 votes. The winner got 4950 more votes than the person who received the least number of votes. Find the total number of votes polled.
33000
Understanding the Election Vote Problem
This problem involves calculating the total number of votes polled in an election contested by three candidates: Chamanlal, Arshad, and Jagit Singh. We are given information about the percentage or share of votes received by some candidates and the difference in votes between the winner and the person who received the least votes. All polled votes are considered valid.
Determining Vote Shares
Let's find the share of votes for each candidate:
- Arshad's Votes: Arshad got 35% of the total votes. If the total number of votes is \(T\), Arshad's votes are \(0.35T\).
- Chamanlal's Votes: We are told that for every 35 votes, Chamanlal got 14 votes. This represents Chamanlal's share of the total votes as a fraction. The fraction is \(\frac{14}{35}\).
Let's simplify Chamanlal's fraction:
\[ \frac{14}{35} = \frac{2 \times 7}{5 \times 7} = \frac{2}{5} \]
To express this as a percentage, we multiply by 100%:
\[ \frac{2}{5} \times 100\% = \frac{200}{5}\% = 40\% \]
So, Chamanlal got 40% of the total votes. Chamanlal's votes are \(0.40T\).
- Jagit Singh's Votes: The total percentage of votes polled is 100%. The sum of percentages for Chamanlal and Arshad is \(40\% + 35\% = 75\%\). The remaining percentage goes to Jagit Singh.
Jagit Singh's percentage = \(100\% - 75\% = 25\%\).
Jagit Singh's votes are \(0.25T\).
Identifying Winner and Loser
Let's compare the percentages to find the winner and the person with the least votes:
- Chamanlal: 40%
- Arshad: 35%
- Jagit Singh: 25%
Clearly, Chamanlal got the highest percentage (40%) and is the winner. Jagit Singh got the lowest percentage (25%) and received the least number of votes.
Using the Vote Difference Information
We are given that the winner got 4950 more votes than the person who received the least number of votes. This difference is between Chamanlal's votes and Jagit Singh's votes.
Difference in votes = Votes of Winner - Votes of Loser = 4950.
In terms of percentages, the difference is:
Difference in percentage = Percentage of Winner - Percentage of Loser = \(40\% - 25\% = 15\%\).
So, 15% of the total votes corresponds to 4950 votes.
We can write this as an equation:
\[ 15\% \text{ of } T = 4950 \]
\[ 0.15 \times T = 4950 \]
Calculating the Total Number of Votes
Now, we need to solve the equation for \(T\):
\[ T = \frac{4950}{0.15} \]
To make the division easier, we can multiply both the numerator and the denominator by 100 to remove the decimal:
\[ T = \frac{4950 \times 100}{0.15 \times 100} = \frac{495000}{15} \]
Now, let's perform the division:
\[ 495000 \div 15 = 33000 \]
So, the total number of votes polled is 33000.
Verification
Let's quickly verify our answer:
- Total votes \(T = 33000\).
- Chamanlal (40%): \(0.40 \times 33000 = 13200\)
- Arshad (35%): \(0.35 \times 33000 = 11550\)
- Jagit Singh (25%): \(0.25 \times 33000 = 8250\)
Sum of votes: \(13200 + 11550 + 8250 = 33000\). This matches the total.
Difference between winner (Chamanlal) and loser (Jagit Singh): \(13200 - 8250 = 4950\). This matches the given difference.
The calculated total number of votes, 33000, is consistent with all the conditions given in the problem.
Revision Table: Election Calculations
| Candidate | Vote Share Information | Percentage of Total Votes | Votes (\(T=33000\)) |
|---|---|---|---|
| Arshad | 35% | 35% | 11550 |
| Chamanlal | 14 out of 35 votes | \(\frac{14}{35} = 40\%\) | 13200 |
| Jagit Singh | Remaining votes | \(100\% - (35\%+40\%) = 25\%\) | 8250 |
Additional Information: Solving Percentage Problems
Percentage problems often involve translating word descriptions into mathematical equations. Key steps include:
- Identifying the total or base value (which might be unknown).
- Converting percentages or ratios into decimal or fractional forms.
- Setting up an equation based on the given relationship (like difference, sum, etc.).
- Solving the equation to find the unknown value.
In this election problem, the difference between the highest and lowest percentages was a direct translation of the given vote difference, allowing us to find the total number of votes.
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