An oil tanker can be emptied by the main pump in 5 hours.  An auxiliary pump can empty the tanker in 9 hours.  If the main pump is started at 7 PM​, when should the auxiliary pump be started so that the tanker is emptied by 11 PM​?

To solve this problem, let's break it down step by step:

### Step 1: Calculate the rate of each pump.
- The main pump can empty the tanker in 5 hours. Therefore, its rate is:
  \[
  \text{Rate of main pump} = \frac{1}{5} \text{ of the tanker per hour}
  \]
- The auxiliary pump can empty the tanker in 9 hours. Therefore, its rate is:
  \[
  \text{Rate of auxiliary pump} = \frac{1}{9} \text{ of the tanker per hour}
  \]

### Step 2: Set up the equation for the time when both pumps are working.
Let \( t \) be the time after 7 PM when the auxiliary pump is started.

- The main pump operates alone from 7 PM to \( t \) hours. In this time, it would have emptied:
  \[
  \text{Tanker emptied by main pump alone} = \frac{t}{5}
  \]
- After \( t \) hours, both pumps work together from \( t \) to 11 PM. Since 11 PM is 4 hours after 7 PM, the time both pumps work together is \( 4 - t \) hours. During this time, they would empty:
  \[
  \text{Tanker emptied by both pumps} = \left( \frac{1}{5} + \frac{1}{9} \right) \times (4 - t)
  \]

### Step 3: Write the equation for the total emptying of the tanker.
Since the tanker is completely emptied by 11 PM, the sum of the above quantities must equal 1 (i.e., the whole tanker):

\[
\frac{t}{5} + \left( \frac{1}{5} + \frac{1}{9} \right) \times (4 - t) = 1
\]

### Step 4: Solve the equation.
First, find the common rate when both pumps are working together:

\[
\frac{1}{5} + \frac{1}{9} = \frac{9 + 5}{45} = \frac{14}{45}
\]

Now, plug this into the equation:

\[
\frac{t}{5} + \frac{14}{45} \times (4 - t) = 1
\]

Multiply the entire equation by 45 to eliminate the fractions:

\[
9t + 14(4 - t) = 45
\]

Expand and simplify:

\[
9t + 56 - 14t = 45
\]

\[
-5t + 56 = 45
\]

Subtract 56 from both sides:

\[
-5t = -11
\]

Divide by -5:

\[
t = \frac{11}{5} = 2.2 \text{ hours}
\]

### Step 5: Convert time to hours and minutes.
2.2 hours is equivalent to 2 hours and 12 minutes.

### Conclusion:
The auxiliary pump should be started **2 hours and 12 minutes after 7 PM**, which is **9:12 PM**.

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Would you like to see more details or have any questions? 

Here are five related questions:
1. How do you calculate the combined work rate of two pumps?
2. What if the auxiliary pump was slower, taking 12 hours instead of 9? When should it be started?
3. How would the solution change if both pumps started together at 7 PM?
4. Can this problem be solved using a graphical approach?
5. How does the time of starting the auxiliary pump affect the total time to empty the tanker?

**Tip:** Always convert time fractions into minutes for practical timekeeping.

Learn how to solve a pump rate problem involving a main pump and an auxiliary pump. Understand how to calculate rates, set up equations, and solve for time. Suitable for high school students and anyone interested in pump rate calculations.

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