A school currently has 8 periods of 45 minutes each every day. If the total working hours of the school remain the same but the number of periods is increased to 9 per day, what will be the duration of each period?

Introduction / Context:
This is a classic time and work style question framed around school periods. It tests the concept that when total work or total time is fixed, changing the number of equal parts changes the size of each part. Here, the total school time per day is fixed, but the number of periods increases, so each period must become shorter.

Given Data / Assumptions:

• Initially there are 8 periods each day.
• Each of these 8 periods is 45 minutes long.
• The total working time of the school per day remains unchanged.
• We change the timetable to have 9 periods in the same total time.
• Each of the 9 periods is assumed to have equal duration.

Concept / Approach:
The total daily school time is the product of the number of periods and the length of each period. Since that total is fixed, we compute it using the original structure and then divide by the new number of periods to get the new duration. This involves basic multiplication and division of time, without any advanced algebra.

Step-by-Step Solution:
Step 1: Compute total school time with the original schedule: 8 periods × 45 minutes per period. Step 2: 8 × 45 = 360 minutes of school time per day. Step 3: With the new schedule, this same 360 minutes will be split into 9 equal periods. Step 4: New period length = total minutes / number of periods = 360 / 9 minutes. Step 5: 360 ÷ 9 = 40 minutes per period. Step 6: So each of the 9 periods must be 40 minutes long to keep the total time unchanged.

Verification / Alternative check:
We can reverse-check the answer. If each of the 9 periods is 40 minutes, then total time = 9 × 40 = 360 minutes. This matches the original total of 8 × 45 = 360 minutes. Since total time is the same in both arrangements, the calculation is verified and consistent with the condition that working hours remain unchanged.

Why Other Options Are Wrong:
30 minutes would give a total of 9 × 30 = 270 minutes, which is too short. 35 minutes would give 9 × 35 = 315 minutes, still less than required. 45 or 50 minutes would result in total times greater than or equal to 405 or 450 minutes, which contradict the fixed total time of 360 minutes. Therefore, only 40 minutes satisfies the constraint.

Common Pitfalls:
Students sometimes try to proportion the change in period length directly with the ratio 8:9 without first computing total time, which can lead to confusion. Others mix minutes and hours incorrectly. The safest approach is always to first compute the total time in minutes, because minutes avoid fractional hours. Then divide by the new number of periods to obtain the new duration.

Final Answer:
Each period will be 40 minutes long.