A team of 8 persons takes part in a shooting competition. The best marksman scores 85 points. If he had scored 92 points instead, the average score of the team would have been 84. What is the actual total number of points scored by the team?

Introduction / Context:
This question tests understanding of how changing a single score affects the average of the whole team. You know the score of the best marksman and how the average would change if he had scored more points. From this hypothetical situation, you must find the actual total score of the team. This is a common type of average and what if scenario in aptitude exams.

Given Data / Assumptions:
- Number of team members = 8.
- Best marksman's actual score = 85 points.
- If he had scored 92 points, the average score of the team would have been 84 points.
- We must find the actual total points scored by the team with the original scores.

Concept / Approach:
The key idea is that increasing one player's score from 85 to 92 increases the team total by 7 points. The hypothetical average of 84 gives us the hypothetical total score, from which we can subtract the 7 extra points to recover the actual total. This method avoids needing to know each individual score and instead uses only total and average relations.

Step-by-Step Solution:
Step 1: If the best marksman had scored 92 points, the average score of the 8 members would have been 84.Step 2: Hypothetical total score with 92 points = 84 * 8 = 672 points.Step 3: The difference between the hypothetical and actual score of the best marksman is 92 - 85 = 7 points.Step 4: So the hypothetical total is 7 points more than the actual total.Step 5: Actual total score of the team = 672 - 7 = 665 points.

Verification / Alternative Check:
If the actual total is 665 and one player has 85, then the sum of the other seven players' scores is 665 - 85 = 580. If that same player had scored 92, the new team total would be 580 + 92 = 672, and the average would be 672 / 8 = 84, matching the given hypothetical situation. This confirms that 665 is the correct total score.

Why Other Options Are Wrong:
Totals such as 657, 658 or 678 do not yield an average of 84 when 7 extra points are added to the best marksman's score. Only 665 satisfies the condition that increasing a single player's score by 7 yields a team total of 672, which corresponds to an average of 84.

Common Pitfalls:
A common mistake is to set up an equation with an unknown average first, which is not necessary here. Another error is to subtract instead of add when translating between actual and hypothetical totals. Remember that changing one score by a certain amount changes the total by the same amount. Work carefully with these differences to avoid sign errors.

Final Answer:
The actual total number of points scored by the team is 665.