Nineteen persons go to a hotel for a combined dinner party. Thirteen of them spend Rs. 79 each on their dinner, and each of the remaining persons spends Rs. 4 more than the average expenditure of all 19 persons. What is the total amount of money spent by the group on dinner?

Introduction / Context:
This problem combines averages with a bit of algebra. Some people in a group spend a fixed amount, while the others spend a certain amount more than the average expenditure of the entire group. We need to use the definition of average and total expenditure to find the overall amount spent on the dinner.

Given Data / Assumptions:

• Total number of persons at the dinner = 19.
• Thirteen persons each spend Rs. 79.
• The remaining 6 persons each spend Rs. 4 more than the average expenditure of all 19 persons.
• Let the average expenditure of all 19 persons be x rupees.
• All expenditures are per person for this one dinner.

Concept / Approach:
If the average expenditure per person is x, then: Total expenditure of all 19 persons = 19 * x. We can also express the total as the sum of the expenditures of the first 13 persons and the remaining 6 persons:

• 13 persons spend 13 * 79.
• 6 persons each spend (x + 4).

Equating these two expressions for total expenditure gives an equation in x, which we can solve to find the average and then the total money spent.

Step-by-Step Solution:
Step 1: Let the average expenditure per person be x rupees. Then total expenditure = 19 * x. Step 2: Express total expenditure using the detailed information. Total expenditure = 13 * 79 + 6 * (x + 4). Step 3: Set up the equation. 19 * x = 13 * 79 + 6 * (x + 4). Step 4: Simplify the right hand side. 13 * 79 = 1027. 6 * (x + 4) = 6 * x + 24. So 19 * x = 1027 + 6 * x + 24 = 1051 + 6 * x. Step 5: Solve for x. 19 * x - 6 * x = 1051. 13 * x = 1051, so x = 1051 / 13 = 80.8461538 rupees (approximately). Step 6: Compute total expenditure. Total expenditure = 19 * x = 19 * 80.8461538 ≈ 1536.08 rupees.

Verification / Alternative check:
We can check by directly computing expenditure with this average. If the average is about Rs. 80.85, the 6 higher spending persons each pay about 84.85 rupees. Then: Total of 13 persons = 13 * 79 = 1027. Total of 6 persons ≈ 6 * 84.85 ≈ 509.1. Grand total ≈ 1027 + 509.1 ≈ 1536.1 rupees, which matches the earlier total within rounding error. Thus the total of about Rs. 1536.08 is consistent.

Why Other Options Are Wrong:
Values like Rs. 1534.00, Rs. 1628.40 and Rs. 1492.00 would correspond to different implied averages that do not satisfy the equation 19 * x = 13 * 79 + 6 * (x + 4). When these totals are substituted back, the per person average and the condition that six persons spent exactly Rs. 4 more than the average fail, so they are incorrect.

Common Pitfalls:
A typical mistake is to assume that the remaining 6 persons each pay Rs. 79 + 4 = Rs. 83 without considering that the extra amount is over the average x, not over Rs. 79. Another error is to forget that total expenditure must be 19 * x and instead try to compute average from partial groups directly. Always set up an algebraic equation equating total from the average with total from individual expenditures.

Final Answer:
The total money spent by all 19 persons is approximately Rs. 1536.08.