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.