Question

A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

• A. 3 hours 15 min
• B. 3 hours 45 min
• C. 3 hours 40 min
• D. 3 hours 50 min

Hint:

Break the work at the switching point. Compute time for each phase using \(\text{time}=\frac{\text{work}}{\text{rate}}\) and add them.

Answer 1

The Correct Option is B

Step 1: Write the single-tap rate. 
One tap fills the tank in 6 hours \(\Rightarrow\) rate \(=\frac{1}{6}\) tank/hour. 

Step 2: Time to fill the first half with one tap. 
\[ t_1=\frac{\text{work}}{\text{rate}}=\frac{\frac12}{\frac16}=3\ \text{hours}. \] 

Step 3: Fill the remaining half with 4 taps (1 existing + 3 new). 
Combined rate \(=4\times\frac{1}{6}=\frac{2}{3}\) tank/hour. 
\[ t_2=\frac{\frac12}{\frac{2}{3}}=\frac{1}{2}\times\frac{3}{2}=\frac{3}{4}\ \text{hour}=45\ \text{minutes}. \] 

Step 4: Total time. 
\[ t_{\text{total}} = t_1+t_2 = 3\ \text{hours}+45\ \text{minutes}=3\ \text{hours }45\ \text{minutes}. \] \[ \boxed{3\ \text{hours }45\ \text{minutes}} \]

Answer 2

Using direct and inverse proportion instead of rate formulas gives another way to reach the same total time.

1. 3 hours 15 min: This equals 195 minutes total; subtracting the 180 minutes (3 hours) needed for one tap to fill the first half leaves only 15 minutes for the second half with 4 taps, but 15 minutes is too short for that portion, so this fails.
2. 3 hours 45 min: This equals 225 minutes; subtracting 180 minutes leaves exactly 45 minutes for the second half, matching the time needed with 4 taps.
3. 3 hours 40 min: This equals 220 minutes; subtracting 180 leaves 40 minutes, which is close but not exactly the 45 minutes required.
4. 3 hours 50 min: This equals 230 minutes; subtracting 180 leaves 50 minutes, more than the 45 minutes actually needed.

One tap fills the whole tank in 6 hours, so by direct proportion it fills half the tank in 3 hours. For the second half, since the work is fixed and the number of taps quadruples from 1 to 4, by inverse proportion the time needed is \(\dfrac{3\ \text{hours}}{4}=45\) minutes. Adding the two stages gives \(3\) hours \(45\) minutes.

So the correct answer is 3 hours 45 min.