One air conditioner can cool a hall in 40 minutes, while another similar air conditioner can cool the same hall in 45 minutes under similar conditions. If both air conditioners are switched on at the same time, approximately how long (in minutes) will it take to cool the hall?

Introduction / Context:
This time and work style question uses the concept of combined rates. Two air conditioners can cool the same room at different speeds. When both are used simultaneously, their cooling rates add up, giving a faster total cooling time. The answer is asked approximately, so a rounded value is acceptable.

Given Data / Assumptions:

• AC1 can cool the hall in 40 minutes.
• AC2 can cool the hall in 45 minutes.
• Both are switched on at the same time to cool the same hall.
• Cooling rates are constant and additive.

Concept / Approach:
Treat cooling the hall as completing one unit of work. Each air conditioner has a work rate of 1 / time. When both operate together, their rates add. The reciprocal of this combined rate gives the total time taken when both work simultaneously. This is analogous to workers finishing a job together.

Step-by-Step Solution:
Let total cooling work = 1 unit. Rate of AC1 = 1 / 40 of the work per minute. Rate of AC2 = 1 / 45 of the work per minute. Combined rate when both run together = 1/40 + 1/45. Compute: 1/40 + 1/45 = (45 + 40) / (40 * 45) = 85 / 1800. Simplify combined rate: 85 / 1800 = 17 / 360 work per minute. Time required = total work / rate = 1 / (17 / 360) = 360 / 17 minutes. Compute 360 / 17 ≈ 21.176 minutes. Rounded to the nearest whole minute, this is approximately 22 minutes.
Verification / Alternative check:
If it were exactly 20 minutes, the amount of work done would be 20 * (1/40 + 1/45) = 20 * 17 / 360 = 340 / 360 ≈ 0.944 of the work, not enough. At 22 minutes, work done = 22 * 17 / 360 = 374 / 360 ≈ 1.039, which is just over full completion. So around 21 to 22 minutes is reasonable, and 22 minutes is the closest given option.
Why Other Options Are Wrong:
18 or 19 minutes are too short; the combined rate would not complete the entire work in such a small time. 24 minutes is longer than necessary, given that 360 / 17 is only about 21.18 minutes.
Common Pitfalls:
A frequent mistake is to average the times (40 and 45) as (40 + 45)/2 = 42.5 minutes, which is conceptually wrong because rates, not times, add. Another error is to compute 1 / ( (40 + 45) ) rather than 1/40 + 1/45. Always work in terms of rates when multiple agents work together.
Final Answer:
The hall will be cooled in approximately 22 minutes.