The following table gives information of panchayat elections held in six villages P, Q, R, S, T and U. Total numberof votes (in hundreds) Votes polled (in %) Valid votes (in %) Hint: 1) percentage of votes polled = \(\rm {{Total \ votes \ polled} \over Total \ number \ of \ votes}\) × 100 2) percentage of valid votes = \(\rm {{Total \ valid \ votes} \over Total \ votes \ polled}\) × 100 Find the ratio of invalid votes of village R to that of village U?Village P 50 80 80 Q 60 75 80 R 100 65 65 S 80 60 70 T 60 80 90 U 40 90 60
455 ∶ 288
Calculating Invalid Votes in Panchayat Elections
This problem asks us to find the ratio of invalid votes between village R and village U based on the provided election data table. We are given the total number of votes, the percentage of votes polled, and the percentage of valid votes (out of the votes polled) for each village.
Here is the data table:
| Village | Total number of votes (in hundreds) | Votes polled (in %) | Valid votes (in %) |
|---|---|---|---|
| P | 50 | 80 | 80 |
| Q | 60 | 75 | 80 |
| R | 100 | 65 | 65 |
| S | 80 | 60 | 70 |
| T | 60 | 80 | 90 |
| U | 40 | 90 | 60 |
To find the number of invalid votes for each village, we need to follow these steps:
- Calculate the total number of votes. The table gives this in hundreds, so multiply by 100.
- Calculate the total votes polled. This is the percentage of total votes that were cast.
- Calculate the total valid votes. This is the percentage of votes polled that were valid.
- Calculate the invalid votes by subtracting the valid votes from the total votes polled.
Calculate Invalid Votes for Village R
- Total number of votes in R: The table shows 100 hundreds, which is \(100 \times 100 = 10000\) votes.
- Votes polled in R: 65% of the total votes were polled. Votes polled = \(\frac{65}{100} \times 10000 = 65 \times 100 = 6500\) votes.
- Valid votes in R: 65% of the votes polled were valid. Valid votes = \(\frac{65}{100} \times 6500\). We can calculate this as \(0.65 \times 6500 = 4225\) votes.
- Invalid votes in R: These are the votes polled minus the valid votes. Invalid votes (R) = Total votes polled - Valid votes Invalid votes (R) = \(6500 - 4225 = 2275\) votes.
Calculate Invalid Votes for Village U
- Total number of votes in U: The table shows 40 hundreds, which is \(40 \times 100 = 4000\) votes.
- Votes polled in U: 90% of the total votes were polled. Votes polled = \(\frac{90}{100} \times 4000 = 90 \times 40 = 3600\) votes.
- Valid votes in U: 60% of the votes polled were valid. Valid votes = \(\frac{60}{100} \times 3600\). We can calculate this as \(0.60 \times 3600 = 2160\) votes.
- Invalid votes in U: These are the votes polled minus the valid votes. Invalid votes (U) = Total votes polled - Valid votes Invalid votes (U) = \(3600 - 2160 = 1440\) votes.
Find the Ratio of Invalid Votes R to U
The question asks for the ratio of invalid votes of village R to that of village U. Ratio = Invalid votes (R) : Invalid votes (U) Ratio = \(2275 : 1440\)
To simplify this ratio, we can divide both numbers by their greatest common divisor. Both numbers end in 5 or 0, so they are divisible by 5.
- \(2275 \div 5 = 455\)
- \(1440 \div 5 = 288\)
The simplified ratio is \(455 : 288\).
Let's check if 455 and 288 have any common factors other than 1. Factors of 455: 1, 5, 7, 13, 35, 65, 91, 455 Factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288 They share no common factors other than 1. Thus, the ratio \(455 : 288\) is in its simplest form.
The ratio of invalid votes of village R to that of village U is \(455 : 288\).
Revision Table: Key Election Calculation Formulas
| Concept | Formula |
|---|---|
| Total Votes Polled | \(\rm (Total \ Number \ of \ Votes) \times \frac{Percentage \ of \ Votes \ Polled}{100}\) |
| Total Valid Votes | \(\rm (Total \ Votes \ Polled) \times \frac{Percentage \ of \ Valid \ Votes}{100}\) |
| Total Invalid Votes | \(\rm Total \ Votes \ Polled - Total \ Valid \ Votes\) |
| Invalid Votes (alternative) | \(\rm (Total \ Votes \ Polled) \times (1 - \frac{Percentage \ of \ Valid \ Votes}{100})\) |
Additional Information on Election Data Analysis
Analyzing election data involves understanding different terms and calculations related to voting. Here are some key points:
- Total Number of Votes: This is the total count of eligible votes in a constituency or area before polling begins.
- Votes Polled: This refers to the actual number of votes that were cast by the voters. It is usually represented as a percentage of the total number of votes. A high percentage indicates high voter turnout.
- Valid Votes: These are the votes that are counted towards the candidates. Votes can be deemed invalid for various reasons, such as improper marking, damage to the ballot paper, or other procedural issues depending on the voting system.
- Invalid Votes: These are the votes that were cast but are not counted towards any candidate. They are the difference between the total votes polled and the total valid votes.
- Voter Turnout: Often expressed as the percentage of registered voters who cast a vote. The "Votes polled (in %)" in the table represents this.
- Calculating Percentages: Remember that percentages are parts per hundred. To convert a percentage to a decimal or fraction for calculation, divide by 100. For example, 65% is \(\frac{65}{100}\) or 0.65.
- Ratio Analysis: Ratios like the one calculated (invalid votes R to U) help compare different values. Ratios can often be simplified by dividing both parts by their greatest common divisor.
Understanding these concepts is crucial for interpreting election statistics and solving problems related to voting data.
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Years
Chandigarh
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Kolkata
2016
1800
2500
1800
2000
2017
2500
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1850
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2018
2300
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2019
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1950
1900
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Schools
Q
R
S
T
A
P
A
P
A
P
A
P
2014
320
240
400
340
420
273
250
225
2015
400
320
380
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350
280
300
228
2016
440
286
360
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264
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256
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350
252
420
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375
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No. of persons
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34
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40
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How many persons earn Rs. 200 or more but less than Rs. 300?
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