An exhibition was conducted for 4 weeks. Compared to the first week, the number of tickets sold in the second week increased by 20%, in the third week increased by 16% over the second week, and in the fourth week decreased by 20% compared to the third week. If 1,392 tickets were sold in the fourth week, how many tickets were sold in the first week?

Introduction / Context:
This question deals with successive percentage changes in the number of tickets sold over four weeks of an exhibition. The sales figures change by different percentages each week relative to the previous week. We are given the final week's sales and asked to work backwards to determine the number of tickets sold originally in the first week. This type of problem reinforces understanding of chained percentage multipliers and reverse calculations.

Given Data / Assumptions:

• Let the number of tickets sold in the first week be T.
• Second week sales = 20% more than first week.
• Third week sales = 16% more than second week.
• Fourth week sales = 20% less than third week.
• Tickets sold in fourth week = 1,392.

Concept / Approach:
Each weekly change can be expressed as a multiplication factor. A 20% increase corresponds to a factor of 1.20, a 16% increase to 1.16, and a 20% decrease to 0.80. If the first week sales are T, the fourth week sales will be T multiplied sequentially by these factors in order. So Fourth week sales = T * 1.20 * 1.16 * 0.80. We then set this equal to 1392 and solve for T to find the initial sales.

Step-by-Step Solution:
Step 1: Let first week tickets = T. Step 2: Second week tickets = T increased by 20% = T * 1.20. Step 3: Third week tickets = second week tickets increased by 16% = T * 1.20 * 1.16. Step 4: Fourth week tickets = third week tickets decreased by 20% = T * 1.20 * 1.16 * 0.80. Step 5: We are given that fourth week tickets = 1392, so T * 1.20 * 1.16 * 0.80 = 1392. Step 6: Compute combined multiplier: 1.20 * 1.16 = 1.392. Step 7: Multiply by 0.80: 1.392 * 0.80 = 1.1136. Step 8: Therefore, 1.1136 * T = 1392. Step 9: Solve for T: T = 1392 / 1.1136. Step 10: Compute T: 1392 / 1.1136 = 1250. Step 11: Hence, 1250 tickets were sold in the first week.

Verification / Alternative check:
Forward check using T = 1250. Second week: 1250 * 1.20 = 1500 tickets. Third week: 1500 * 1.16 = 1740 tickets. Fourth week: 1740 * 0.80 = 1392 tickets. This matches the given fourth week sales exactly, confirming that the initial week sales were correctly computed as 1250 tickets.

Why Other Options Are Wrong:
If we assume values like 1124, 1420, 1345, or 1300 and apply the same sequence of percentage changes, the resulting fourth week sales will not equal 1392. They will produce either lower or higher values. Only T = 1250, when multiplied by the combined factor 1.1136, gives exactly 1392, so the other options are inconsistent with the described percentage changes.

Common Pitfalls:
A common error is to treat all percentage changes as relative to the original week instead of successive changes week by week. Another mistake is to add or subtract percentages directly instead of using multiplication factors. Some learners also misapply the 20% decrease by subtracting from the wrong base. Working stepwise and carefully applying each increase or decrease factor in order prevents these problems.

Final Answer:
The number of tickets sold in the first week was 1,250.