Question

One AC cools in 40 minutes and another in 45 minutes. If both work together, how long will it take?

• A. 18 minutes
• B. 19 minutes
• C. 22 minutes
• D. 24 minutes

Hint:

For work-and-time problems, always convert individual times into rates (work per unit time). Sum the rates for combined work, then take the reciprocal for the total time. Be careful with calculations and comparing to options, as rounding or question errors can occur.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem involves two ACs (working units) completing a task (cooling) at different rates. We need to find the combined time it takes for them to complete the same task when working together. This is a classic "work and time" problem.

Step 2: Key Formula or Approach:
1. Determine the rate of work for each AC (amount of work done per unit time). If an AC cools in $T$ minutes, its rate is $1/T$ per minute.
2. Add their individual rates to find the combined rate.
3. The inverse of the combined rate will give the total time taken when working together.
If time taken by A is $T_1$ and by B is $T_2$, then time taken together $T = \frac{T_1 \times T_2}{T_1 + T_2}$.

Step 3: Detailed Explanation:
Let AC1 cool in $T_1 = 40$ minutes.
Let AC2 cool in $T_2 = 45$ minutes.
1. Rate of AC1:
Rate1 = $\frac{1}{40}$ of the cooling per minute.
2. Rate of AC2:
Rate2 = $\frac{1}{45}$ of the cooling per minute.
3. Combined Rate:
Combined Rate = Rate1 + Rate2 = $\frac{1}{40} + \frac{1}{45}$.
To add these fractions, find a common denominator, which is the Least Common Multiple (LCM) of 40 and 45.
$40 = 2^3 \times 5$
$45 = 3^2 \times 5$
LCM(40, 45) = $2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$.
Combined Rate = $\frac{9}{360} + \frac{8}{360} = \frac{17}{360}$ of the cooling per minute.
4. Time taken together (T):
Time = $\frac{1}{\text{Combined Rate}} = \frac{360}{17}$ minutes.
$360 \div 17 \approx 21.176$ minutes \approx 22$ minutes

Step 4: Final Answer:
When both ACs work together, it will take approximately 22 minutes.

Answer 2

Instead of adding the two rates directly, we can test each candidate time by checking whether the reciprocal of that time actually equals the sum of the two individual cooling rates, \( \frac{1}{40} + \frac{1}{45} \).

1. 18 minutes: If the combined time were 18 minutes, the combined rate would need to be \( \frac{1}{18} \approx 0.0556 \) of the job per minute. The actual combined rate is \( \frac{1}{40} + \frac{1}{45} = \frac{9}{360} + \frac{8}{360} = \frac{17}{360} \approx 0.0472 \), noticeably smaller than \( 0.0556 \), so this time is too short to be correct.
2. 19 minutes: A combined time of 19 minutes would require a rate of \( \frac{1}{19} \approx 0.0526 \), which is still higher than the actual combined rate of \( 0.0472 \), so this option is also too fast.
3. 22 minutes: A combined time of 22 minutes corresponds to a rate of \( \frac{1}{22} \approx 0.0455 \), very close to the true combined rate of \( \frac{17}{360} \approx 0.0472 \); since \( \frac{360}{17} \approx 21.18 \) minutes rounds up to 22 minutes as the practical time needed to finish the job, this option matches.
4. 24 minutes: A combined time of 24 minutes implies a rate of \( \frac{1}{24} \approx 0.0417 \), which is smaller than the true combined rate of \( 0.0472 \), meaning this option overestimates how long the two ACs would actually take together, so it is too slow.

Comparing each candidate time's implied rate against the true combined cooling rate shows that only the time closest to \( \frac{360}{17} \) minutes is consistent with both ACs working together.

Therefore, the correct answer is 22 minutes.