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Sagot :
Sure, let's break down the problem step by step.
1. Percentage Representation:
- Amar received [tex]\( 33 \frac{1}{3} \% \)[/tex] of the total votes.
This can be rewritten as a fraction:
[tex]\[ 33 \frac{1}{3} \% = \frac{100}{3} \% = \frac{100}{3} \times \frac{1}{100} = \frac{1}{3} \][/tex]
- Bhima received [tex]\( 42 \frac{6}{7} \% \)[/tex] of the remaining votes after Amar's share.
This can be rewritten as:
[tex]\[ 42 \frac{6}{7} \% = \frac{300}{7} \% = \frac{300}{7} \times \frac{1}{100} \][/tex]
2. Unknowns and Basic Equations:
- Let [tex]\( V \)[/tex] be the total number of votes polled.
- Amar received: [tex]\( \frac{1}{3}V \)[/tex] votes.
- Remaining votes after Amar's share: [tex]\( V - \frac{1}{3}V = \frac{2}{3}V \)[/tex].
3. Bhima's Votes:
- Bhima received [tex]\( \frac{300}{7 \times 100} \)[/tex] of the remaining votes:
[tex]\[ \text{Bhima's votes} = \frac{300}{700} \times \frac{2}{3}V = \frac{300}{7} \times \frac{1}{100} \times \frac{2}{3}V = \frac{600}{21} \times V = \frac{200}{7} \times V \][/tex]
4. Relation Given:
- Amar got 4000 votes more than Bhima:
[tex]\[ \frac{1}{3}V = \frac{200}{7}V + 4000 \][/tex]
5. Solving the Equation:
- Let's solve for [tex]\( V \)[/tex]:
[tex]\[ \frac{1}{3}V - \frac{200}{7}V = 4000 \][/tex]
- Convert fractions to have a common denominator (LCM of 3 and 7 is 21):
[tex]\[ \frac{7}{21}V - \frac{600}{21}V = 4000 \][/tex]
- Combine the fractions:
[tex]\[ \frac{7 - 600}{21}V = 4000 \][/tex]
[tex]\[ -\frac{593}{21}V = 4000 \][/tex]
[tex]\[ V = \frac{4000 \times 21}{593} \][/tex]
- Simplifying the fraction:
[tex]\[ V \approx 84084.08408408421 \][/tex]
Thus, the total number of votes polled is approximately [tex]\( 84084 \)[/tex].
6. Verification:
- Amar's votes:
[tex]\[ \frac{1}{3} \times 84084.084084 \approx 28025.225225225266 \][/tex]
- Remaining votes after Amar:
[tex]\[ \frac{2}{3} \times 84084.084084 \approx 56056.056056 \][/tex]
- Bhima's votes:
[tex]\[ \frac{42 \frac{6}{7} \%}{100 \%} \times 56056.056056 \approx 24025.225225225262 \][/tex]
- Difference between Amar's and Bhima's votes:
[tex]\[ 28025.225225 - 24025.225225 \approx 4000 \][/tex]
Thus, all the calculations confirm the result, and the total number of votes polled is [tex]\( 84084 \)[/tex].
1. Percentage Representation:
- Amar received [tex]\( 33 \frac{1}{3} \% \)[/tex] of the total votes.
This can be rewritten as a fraction:
[tex]\[ 33 \frac{1}{3} \% = \frac{100}{3} \% = \frac{100}{3} \times \frac{1}{100} = \frac{1}{3} \][/tex]
- Bhima received [tex]\( 42 \frac{6}{7} \% \)[/tex] of the remaining votes after Amar's share.
This can be rewritten as:
[tex]\[ 42 \frac{6}{7} \% = \frac{300}{7} \% = \frac{300}{7} \times \frac{1}{100} \][/tex]
2. Unknowns and Basic Equations:
- Let [tex]\( V \)[/tex] be the total number of votes polled.
- Amar received: [tex]\( \frac{1}{3}V \)[/tex] votes.
- Remaining votes after Amar's share: [tex]\( V - \frac{1}{3}V = \frac{2}{3}V \)[/tex].
3. Bhima's Votes:
- Bhima received [tex]\( \frac{300}{7 \times 100} \)[/tex] of the remaining votes:
[tex]\[ \text{Bhima's votes} = \frac{300}{700} \times \frac{2}{3}V = \frac{300}{7} \times \frac{1}{100} \times \frac{2}{3}V = \frac{600}{21} \times V = \frac{200}{7} \times V \][/tex]
4. Relation Given:
- Amar got 4000 votes more than Bhima:
[tex]\[ \frac{1}{3}V = \frac{200}{7}V + 4000 \][/tex]
5. Solving the Equation:
- Let's solve for [tex]\( V \)[/tex]:
[tex]\[ \frac{1}{3}V - \frac{200}{7}V = 4000 \][/tex]
- Convert fractions to have a common denominator (LCM of 3 and 7 is 21):
[tex]\[ \frac{7}{21}V - \frac{600}{21}V = 4000 \][/tex]
- Combine the fractions:
[tex]\[ \frac{7 - 600}{21}V = 4000 \][/tex]
[tex]\[ -\frac{593}{21}V = 4000 \][/tex]
[tex]\[ V = \frac{4000 \times 21}{593} \][/tex]
- Simplifying the fraction:
[tex]\[ V \approx 84084.08408408421 \][/tex]
Thus, the total number of votes polled is approximately [tex]\( 84084 \)[/tex].
6. Verification:
- Amar's votes:
[tex]\[ \frac{1}{3} \times 84084.084084 \approx 28025.225225225266 \][/tex]
- Remaining votes after Amar:
[tex]\[ \frac{2}{3} \times 84084.084084 \approx 56056.056056 \][/tex]
- Bhima's votes:
[tex]\[ \frac{42 \frac{6}{7} \%}{100 \%} \times 56056.056056 \approx 24025.225225225262 \][/tex]
- Difference between Amar's and Bhima's votes:
[tex]\[ 28025.225225 - 24025.225225 \approx 4000 \][/tex]
Thus, all the calculations confirm the result, and the total number of votes polled is [tex]\( 84084 \)[/tex].
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