In an election with only two candidates, a candidate secures 70% of the valid votes and wins by a majority of 172 votes. What is the total number of valid votes?

Introduction / Context:
This is another election based percentage question, focusing on valid votes. Two candidates contest, and one candidate receives 70% of the valid votes. The margin of victory in votes is given, and we are asked to find the total number of valid votes. This tests the understanding that the sum of both candidates' vote shares is 100% and that the majority is the difference between their vote counts.

Given Data / Assumptions:

There are exactly two candidates in the election.

The winning candidate secures 70% of the valid votes.

The losing candidate receives the remaining 30% of the valid votes.

The majority (difference between their vote counts) is 172 votes.

We must determine the total number of valid votes.

Concept / Approach:
If the total valid votes are V, the winner gets 0.70V and the loser gets 0.30V. The majority is the difference 0.70V - 0.30V = 0.40V. This difference is equal to the given margin of 172 votes. So we set up the equation 0.40V = 172 and solve for V. This is a direct proportionality relationship, very similar in structure to other percentage difference problems.

Step-by-Step Solution:
Let the total number of valid votes be V.Votes for winner = 70% of V = 0.70V.Votes for loser = 30% of V = 0.30V.Majority = votes for winner minus votes for loser.Majority = 0.70V - 0.30V = 0.40V.Given majority = 172 votes.Hence 0.40V = 172.So V = 172 / 0.40.Compute: 172 / 0.40 = 1720 / 4 = 430.Thus, total number of valid votes is 430.

Verification / Alternative check:
Let us verify with V = 430. Winner's votes = 70% of 430 = 0.70 * 430 = 301. Loser's votes = 30% of 430 = 0.30 * 430 = 129. The difference between them is 301 - 129 = 172 votes, which exactly matches the given majority. Therefore, 430 valid votes is the correct total.

Why Other Options Are Wrong:

446 would give majority = 40% of 446 = 178.4, which is not an integer and not equal to 172.

415 would give majority = 40% of 415 = 166, different from the required margin.

404 would give majority = 40% of 404 = 161.6, again not 172.

400 would give majority = 40% of 400 = 160, which is also incorrect.

Common Pitfalls:
Some learners mistakenly calculate 70% of V as the majority itself instead of the winner's total. Others forget that the loser receives 30% and do not compute the difference correctly. A few may also misinterpret the majority as 70% minus 50% or make arithmetic errors while dividing 172 by 0.40. Always express both candidates' vote counts as percentages of the same base and then form a difference equation with the given majority.

Final Answer:
The total number of valid votes cast in the election is 430.