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

Introduction / Context:
This problem involves averages and a hypothetical change in one person's score. You are told what the team average would have been if the best marksman had scored slightly more. Using this, you can deduce the team's actual total score before that hypothetical change.

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

Concept / Approach:
The key idea is to use the hypothetical scenario to compute the team total if the best marksman had scored 92. Then we recognize that this hypothetical total is exactly 7 points more than the real total, because the only difference is his score changing from 85 to 92. Subtracting 7 from the hypothetical total gives the actual total score.

Step-by-Step Solution:
Step 1: Let T be the actual total score of the team with the best marksman scoring 85.Step 2: If he had scored 92 instead of 85, his score would increase by 7 points.Step 3: Then the hypothetical total score would be T + 7.Step 4: In that hypothetical case, the average score per person would be 84.Step 5: Therefore, (T + 7) / 8 = 84.Step 6: Multiply both sides by 8: T + 7 = 84 * 8.Step 7: Compute 84 * 8 = 672.Step 8: So T + 7 = 672, which gives T = 672 - 7 = 665.

Verification / Alternative check:
With T = 665, the original average is 665 / 8 = 83.125, which is slightly below 84. If we increase only one person's score by 7 points, new total becomes 665 + 7 = 672. New average = 672 / 8 = 84, exactly as stated in the problem. Thus, the total score of 665 is consistent with the hypothetical condition.

Why Other Options Are Wrong:
672: This is the hypothetical total after adding 7 points, not the actual total.645, 588: These totals do not satisfy the average condition when 7 is added and divided by 8. The average does not become 84 in those cases.

Common Pitfalls:
Students sometimes confuse the actual and hypothetical totals, directly using 672 as the answer. Another issue is forgetting that only one person's score changes, so the difference is exactly 7 points. Always distinguish between the real scenario and the imagined one used for calculation.

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