In an election between two candidates, the defeated candidate secured 42% of the valid votes polled and lost the election by 7,840 votes. If 82,560 votes were declared invalid and... In an election between two candidates, the defeated candidate secured 42% of the valid votes polled and lost the election by 7,840 votes. If 82,560 votes were declared invalid and 20% people did NOT cast their vote, then the invalid votes were what percentage (rounded off to 1 decimal place) of the votes which people did NOT cast?
Understand the Problem
The question is asking for the percentage of invalid votes compared to the total votes that people did not cast in an election. It involves calculating the total number of votes and then determining the proportion of invalid votes based on the information given.
Answer
$10.6\%$
Answer for screen readers
The percentage of invalid votes compared to the total votes that people did not cast is ( 10.6% ).
Steps to Solve
- Find Total Valid Votes
Let the total valid votes be ( V ). Since the defeated candidate secured 42% of the valid votes and lost by 7,840 votes, we can set up the equation: [ 0.42V + 7840 = \text{Votes for the winning candidate} ] Let the votes for the winning candidate be ( W ). Rearranging gives: [ W = 0.42V + 7840 ]
- Calculate Total Votes
We know that total votes consist of valid votes plus invalid votes plus those not cast: [ \text{Total Votes} = V + 82,560 + 0.20 \times \text{Total Votes} ] Let ( T ) represent total votes. Rearranging gives: [ T = V + 82,560 + 0.20T ] This simplifies to: [ 0.80T = V + 82,560 ] [ T = \frac{V + 82,560}{0.80} ]
- Express Valid Votes in Terms of Total Votes
Using the definition of valid votes: [ V = T - 82,560 - 0.20T ] This gives: [ V = 0.80T - 82,560 ]
- Combine Equations to Solve for ( V )
Now substitute ( V ) from the second equation into the first: [ 0.80T - 82,560 = \frac{T - 82,560}{0.20} + 7840 ] We can solve this equation to find ( V ).
- Calculate Invalid Votes as a Percentage of Non-Cast Votes
From the problem statement, 20% of total votes are those who did not cast their vote: [ \text{Non-Cast Votes} = 0.20T ] Next, invalid votes are given as ( 82,560 ). The required percentage of invalid votes compared to non-cast votes is: [ \text{Percentage} = \left( \frac{82,560}{0.20T} \right) \times 100 ]
- Substituting and Rounding
After calculating ( T ), substitute back to find the final percentage rounded to one decimal place.
The percentage of invalid votes compared to the total votes that people did not cast is ( 10.6% ).
More Information
This problem illustrates how to use basic algebra to solve for unknowns in an election context, particularly working with percentages and equations involving valid, invalid, and non-cast votes.
Tips
- Misunderstanding the Total Votes: Confusing total votes with just valid votes. Remember to account for both invalid and non-cast votes.
- Percentage Calculations: Failing to convert the fraction into a percentage correctly. Always multiply by 100 after finding the fraction.
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