Working alone at its constant rate, pump X pumped out 1/4 of the water in a tank in 2 hours....

1. Translate the problem requirements: Pump X removes \(\frac{1}{4}\) of water in 2 hours. Then X, Y, Z together remove remaining \(\frac{3}{4}\) in 3 hours. Y alone would take 18 hours for that \(\frac{3}{4}\). Find how long Z alone takes for ALL the water.
2. Calculate individual pump rates: Determine each pump's rate in terms of 'fraction of total tank per hour'
3. Set up the combined work equation: Use the fact that when pumps work together, their rates add up
4. Solve for pump Z's rate: Use the combined equation to find Z's rate, then calculate time for full tank

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

• Pump X works alone and removes \(\frac{1}{4}\) of all the water in 2 hours
• Then all three pumps (X, Y, and Z) work together and remove the remaining \(\frac{3}{4}\) of the water in 3 hours
• We're told that pump Y alone would have needed 18 hours to remove that same \(\frac{3}{4}\) of water
• We need to find how long pump Z alone would take to remove ALL the water in the tank

Think of this like three people working together to empty a swimming pool with buckets. Each person has their own steady pace (rate), and when they work together, their efforts combine.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships

2. Calculate individual pump rates

Let's figure out how much work each pump does per hour. We'll measure this as "fraction of the total tank per hour."

Pump X's rate:
Pump X removed \(\frac{1}{4}\) of the tank in 2 hours.
So pump X's rate = \(\frac{\frac{1}{4} \text{ tank}}{2 \text{ hours}} = \frac{1}{8}\) of the tank per hour

Pump Y's rate:
Pump Y alone would take 18 hours to remove \(\frac{3}{4}\) of the tank.
So pump Y's rate = \(\frac{\frac{3}{4} \text{ tank}}{18 \text{ hours}} = \frac{3}{72} = \frac{1}{24}\) of the tank per hour

Now we know:
• Pump X removes \(\frac{1}{8}\) of the tank per hour
• Pump Y removes \(\frac{1}{24}\) of the tank per hour
• Pump Z removes ? of the tank per hour (this is what we need to find)

3. Set up the combined work equation

When pumps work together, their rates add up - just like when multiple people are filling buckets, the total water moved per hour is the sum of what each person moves.

We know that pumps X, Y, and Z working together removed \(\frac{3}{4}\) of the tank in 3 hours.

Combined rate of all three pumps = \(\frac{\frac{3}{4} \text{ tank}}{3 \text{ hours}} = \frac{1}{4}\) of the tank per hour

Since rates add when working together:
Pump X rate + Pump Y rate + Pump Z rate = Combined rate
\(\frac{1}{8} + \frac{1}{24} + \text{Pump Z rate} = \frac{1}{4}\)

4. Solve for pump Z's rate

Now we solve for pump Z's rate:

\(\frac{1}{8} + \frac{1}{24} + \text{Pump Z rate} = \frac{1}{4}\)

First, let's find \(\frac{1}{8} + \frac{1}{24}\). Using 24 as our common denominator:
\(\frac{1}{8} = \frac{3}{24}\)
So: \(\frac{3}{24} + \frac{1}{24} = \frac{4}{24} = \frac{1}{6}\)

Now our equation becomes:
\(\frac{1}{6} + \text{Pump Z rate} = \frac{1}{4}\)
\(\text{Pump Z rate} = \frac{1}{4} - \frac{1}{6}\)

Using 12 as our common denominator:
\(\frac{1}{4} = \frac{3}{12}\) and \(\frac{1}{6} = \frac{2}{12}\)
So: \(\text{Pump Z rate} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}\) of the tank per hour

Finding the time for the full tank:
If pump Z removes \(\frac{1}{12}\) of the tank per hour, then to remove the entire tank:
Time = \(1 \text{ tank} \div \left(\frac{1}{12} \text{ tank per hour}\right) = 12 \text{ hours}\)

5. Final Answer

Pump Z, working alone at its constant rate, would take 12 hours to pump out all of the water that was pumped out of the tank.

This matches answer choice (B) 12.

Verification: Let's check our work by confirming the combined rate:
• Pump X: \(\frac{1}{8}\) tank per hour
• Pump Y: \(\frac{1}{24}\) tank per hour
• Pump Z: \(\frac{1}{12}\) tank per hour
• Combined: \(\frac{3}{24} + \frac{1}{24} + \frac{2}{24} = \frac{6}{24} = \frac{1}{4}\) tank per hour ✓
In 3 hours, they would remove \(3 \times \frac{1}{4} = \frac{3}{4}\) of the tank ✓

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting what pump Y's time refers to
Students often confuse what "the rest of the water" means when told that pump Y would take 18 hours to pump it out. They might think this refers to the entire tank, when it actually refers only to the remaining \(\frac{3}{4}\) of the tank that was left after pump X finished. This leads to calculating pump Y's rate incorrectly as \(\frac{1}{18}\) of the total tank per hour instead of the correct \(\frac{\frac{3}{4}}{18} = \frac{1}{24}\) of the tank per hour.

Faltering Point 2: Confusion about what the final question is asking
Students may miss that the question asks for the time pump Z would take to pump out "all of the water that was pumped out of the tank" (meaning the entire tank), not just the remaining \(\frac{3}{4}\) that pump Z helped with. This misinterpretation would lead them to find how long pump Z takes to pump \(\frac{3}{4}\) of the tank instead of the full tank.

Faltering Point 3: Setting up incorrect work rate relationships
Students might incorrectly assume that since the three pumps worked together for 3 hours to finish \(\frac{3}{4}\) of the tank, each pump contributed equally to this work. This false assumption bypasses the need to use individual rates and leads to incorrect calculations where they divide the work equally among the three pumps.

Errors while executing the approach

Faltering Point 1: Fraction arithmetic errors when finding common denominators
When calculating \(\frac{1}{8} + \frac{1}{24}\) or \(\frac{1}{4} - \frac{1}{6}\), students frequently make errors with finding common denominators or adding/subtracting fractions. For example, they might incorrectly calculate \(\frac{1}{8} + \frac{1}{24}\) as \(\frac{2}{32}\) instead of the correct \(\frac{1}{6}\), or miscalculate \(\frac{1}{4} - \frac{1}{6}\) as \(\frac{1}{2}\) instead of \(\frac{1}{12}\).

Faltering Point 2: Incorrect conversion between rates and times
Students often confuse the relationship between rate and time. If they correctly find that pump Z's rate is \(\frac{1}{12}\) of the tank per hour, they might incorrectly conclude that it takes \(\frac{1}{12}\) hours to empty the tank, rather than understanding that time = work ÷ rate = \(1 \div \left(\frac{1}{12}\right) = 12\) hours.

Faltering Point 3: Sign errors in the combined rate equation
When setting up the equation (pump X rate) + (pump Y rate) + (pump Z rate) = (combined rate), students might accidentally subtract instead of add rates, or incorrectly rearrange the equation when solving for pump Z's rate, leading to wrong values.

Errors while selecting the answer

Faltering Point 1: Selecting the reciprocal of the correct answer
If students correctly calculate that pump Z's rate is \(\frac{1}{12}\) tank per hour, they might mistakenly select \(\frac{1}{12}\) as their final answer if it were among the choices, rather than recognizing that the question asks for time (12 hours), not the rate fraction.

Faltering Point 2: Confusing partial tank time with full tank time
Students who misunderstood the final question might correctly calculate how long pump Z takes to pump \(\frac{3}{4}\) of the tank (which would be \(\frac{3}{4} \times 12 = 9\) hours) but then look for the closest answer choice rather than recognizing their conceptual error about what the question was asking.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient total tank size

Let's say the tank holds 36 gallons total (chosen because 36 is divisible by 4, 3, and 18, making our calculations clean).

Step 2: Calculate what pump X accomplished

Pump X removed \(\frac{1}{4}\) of the tank in 2 hours = \(\frac{1}{4} \times 36 = 9\) gallons in 2 hours

So pump X's rate = 9 gallons ÷ 2 hours = 4.5 gallons per hour

Step 3: Determine the remaining work

Water left after pump X worked alone = 36 - 9 = 27 gallons

This remaining 27 gallons was pumped out by all three pumps (X, Y, Z) working together in 3 hours.

Step 4: Calculate pump Y's rate

Pump Y alone would take 18 hours to pump the remaining 27 gallons

So pump Y's rate = 27 gallons ÷ 18 hours = 1.5 gallons per hour

Step 5: Find the combined rate of all three pumps

All three pumps together removed 27 gallons in 3 hours

Combined rate = 27 gallons ÷ 3 hours = 9 gallons per hour

Step 6: Calculate pump Z's rate

Combined rate = Pump X rate + Pump Y rate + Pump Z rate

9 = 4.5 + 1.5 + Pump Z rate

9 = 6 + Pump Z rate

Pump Z rate = 3 gallons per hour

Step 7: Find time for pump Z to empty the entire tank

Time for pump Z alone to pump all 36 gallons = 36 gallons ÷ 3 gallons per hour = 12 hours

Answer: (B) 12