Percentage of total voters captured by two political candidates

Question

In an election, 2.8 million votes were cast and each vote was either for candidate I or candidate II. candidate I received 28,000 more votes than candidate II. What percent of the 2,8 million votes were cast for candidate I?

I solved that candidate I received 1% more vote than candidate two.

Then I incorrectly concluded that because candidate I recived 1 % more vote than candidate 2, candidate one would have 50.1% of the total votes.

Why does candidate I actually have 50.5% of the total votes and not 51% of the total votes?

I don't want an algebraic answer.

Answer 1

The reason why it is not $51\%$ is that if one candidate $51\%,$ only $49\%$ can be cast for the other candidate. And $51 - 49 = 2,$ which is larger than the percentage difference between the candidate.

Imagine that exactly $50\%$ of all voters intended to vote for each candidate, but at the last moment some of the candidate-II voters changed their minds and voted for candidate I instead. Each voter who changed his or her mind added one vote to candidate I and took away one vote from candidate II, so that one voter is responsible for creating a two-vote difference between the candidates. Now repeat that $14000$ times ($\frac 12 \%$ of all the votes), and the difference will be $28000$ votes.

Answer 2

Why does candidate I actually have 50.5% of the total votes and not 51% of the total votes?

Because $50.5\text{%}-49.5\text{%}=1\text{%}$ and $50.5\text{%}+49.5\text{%}=100\text{%}$

Edit:

To figure out the percentage that candidate I is above 50% and candidate II is under 50% the equation is

$(50\text{%}+x)-(50\text{%}-x)=1\text{%}$

Answer 3

If each candidate had the same number of votes, then each would have $1,400,000$ votes. Since candidate I received $28,000$ more votes than candidate II, candidate I received $$1,400,000 + 14,000 = 1,414,000$$ votes while candidate II received $$1,400,000 - 14,0000 = 1,386,000$$ votes. Thus, candidate I received

$$\frac{1,414,000}{2,800,000} \cdot 100\% = 50.5\%$$

of the votes while candidate II received $$100\% - 50.5\% = 49.5\%$$ of the votes.