Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a...

1. Translate the problem requirements: We have three pumps with individual rates. When pairs work together, they complete the job in specific times: A+B in \(\frac{6}{5}\) hours, A+C in \(\frac{3}{2}\) hours, B+C in 2 hours. We need to find how long all three take working together.
2. Set up rate equations using the work principle: Express each pump's rate as fraction of tank filled per hour, then use the fact that combined rates equal the sum of individual rates.
3. Create a system from the given pair combinations: Convert the given completion times into rate equations, where rate = 1/time for each pump pair.
4. Solve for the combined rate of all three pumps: Use algebraic manipulation to find (A + B + C) rate directly without finding individual rates first.
5. Convert combined rate to time: Since rate = 1/time, the time for all three pumps equals 1/(combined rate).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with in everyday terms. Think of each pump as having its own "strength" - how much of the tank it can fill in one hour. When pumps work together, their strengths add up.

Here's what we know:

• Pumps A and B together can fill the whole tank in \(\frac{6}{5}\) hours (that's 1.2 hours)
• Pumps A and C together can fill the whole tank in \(\frac{3}{2}\) hours (that's 1.5 hours)
• Pumps B and C together can fill the whole tank in 2 hours

We want to find: How long does it take when all three pumps A, B, and C work together?

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

2. Set up rate equations using the work principle

Now let's think about "rates." A rate tells us how much work gets done per hour. If a pump can fill the whole tank in 2 hours, then in 1 hour it fills \(\frac{1}{2}\) of the tank.

Let's call the rates:

• Rate of pump A = a (fraction of tank filled per hour)
• Rate of pump B = b (fraction of tank filled per hour)
• Rate of pump C = c (fraction of tank filled per hour)

When pumps work together, their rates simply add up. So if pump A fills \(\frac{1}{3}\) of a tank per hour and pump B fills \(\frac{1}{4}\) of a tank per hour, together they fill \(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\) of the tank per hour.

3. Create a system from the given pair combinations

Now we can translate each piece of information into rate equations. Remember: if it takes t hours to complete a job, the rate is \(\frac{1}{t}\) jobs per hour.

• A and B together take \(\frac{6}{5}\) hours, so their combined rate is \(\frac{1}{\frac{6}{5}} = \frac{5}{6}\) tanks per hour
  This gives us: \(\mathrm{a} + \mathrm{b} = \frac{5}{6}\)
• A and C together take \(\frac{3}{2}\) hours, so their combined rate is \(\frac{1}{\frac{3}{2}} = \frac{2}{3}\) tanks per hour
  This gives us: \(\mathrm{a} + \mathrm{c} = \frac{2}{3}\)
• B and C together take 2 hours, so their combined rate is \(\frac{1}{2}\) tanks per hour
  This gives us: \(\mathrm{b} + \mathrm{c} = \frac{1}{2}\)

So our system is:
\(\mathrm{a} + \mathrm{b} = \frac{5}{6}\)
\(\mathrm{a} + \mathrm{c} = \frac{2}{3}\)
\(\mathrm{b} + \mathrm{c} = \frac{1}{2}\)

4. Solve for the combined rate of all three pumps

Here's a clever insight: we want to find \(\mathrm{a} + \mathrm{b} + \mathrm{c}\) (the combined rate of all three pumps), and we don't actually need to find a, b, and c individually!

Let's add all three equations together:
\((\mathrm{a} + \mathrm{b}) + (\mathrm{a} + \mathrm{c}) + (\mathrm{b} + \mathrm{c}) = \frac{5}{6} + \frac{2}{3} + \frac{1}{2}\)

The left side becomes: \(2\mathrm{a} + 2\mathrm{b} + 2\mathrm{c} = 2(\mathrm{a} + \mathrm{b} + \mathrm{c})\)

For the right side, let's find a common denominator. The LCD of 6, 3, and 2 is 6:
\(\frac{5}{6} + \frac{2}{3} + \frac{1}{2} = \frac{5}{6} + \frac{4}{6} + \frac{3}{6} = \frac{12}{6} = 2\)

So: \(2(\mathrm{a} + \mathrm{b} + \mathrm{c}) = 2\)
Therefore: \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 1\)

This means all three pumps together have a rate of 1 tank per hour!

Process Skill: MANIPULATE - Using algebraic techniques to avoid solving for individual variables

5. Convert combined rate to time

If the combined rate is 1 tank per hour, this means all three pumps working together can fill exactly 1 complete tank in 1 hour.

Since \(\mathrm{rate} = \frac{1}{\mathrm{time}}\), we have: \(\mathrm{time} = \frac{1}{\mathrm{rate}} = \frac{1}{1} = 1\) hour

Final Answer

Pumps A, B, and C operating simultaneously can fill the tank in 1 hour.

Looking at our answer choices, this corresponds to choice E: 1.

We can verify this makes sense: each pair of pumps takes between 1.2 and 2 hours, so having all three pumps working together should definitely be faster than any pair, and 1 hour fits this expectation perfectly.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting rate vs. time relationships

Students often confuse the relationship between rate and time. When told "pumps A and B can fill the tank in \(\frac{6}{5}\) hours," they might incorrectly think this means their combined rate is \(\frac{6}{5}\) tanks per hour, rather than understanding that \(\mathrm{rate} = \frac{1}{\mathrm{time}}\), so the rate is actually \(\frac{5}{6}\) tanks per hour.

2. Setting up incorrect rate equations

Students may struggle with the concept that when pumps work together, their rates add up. They might try to multiply rates instead of adding them, or set up equations like "\(\mathrm{a} \times \mathrm{b} = \frac{5}{6}\)" instead of "\(\mathrm{a} + \mathrm{b} = \frac{5}{6}\)".

3. Attempting to solve for individual rates unnecessarily

Many students will immediately try to solve the system of three equations to find the individual values of a, b, and c, not realizing they can directly find \(\mathrm{a} + \mathrm{b} + \mathrm{c}\) by adding all three equations together. This leads to more complex algebra and potential errors.

Errors while executing the approach

1. Common denominator calculation errors

When adding fractions \(\frac{5}{6} + \frac{2}{3} + \frac{1}{2}\), students frequently make arithmetic mistakes. They might use the wrong least common denominator (using 12 instead of 6), or incorrectly convert fractions (writing \(\frac{2}{3}\) as \(\frac{3}{6}\) instead of \(\frac{4}{6}\)).

2. Algebraic manipulation errors

When expanding \((\mathrm{a} + \mathrm{b}) + (\mathrm{a} + \mathrm{c}) + (\mathrm{b} + \mathrm{c})\), students may incorrectly get "\(\mathrm{a} + \mathrm{b} + \mathrm{c}\)" instead of "\(2\mathrm{a} + 2\mathrm{b} + 2\mathrm{c} = 2(\mathrm{a} + \mathrm{b} + \mathrm{c})\)". This leads them to conclude that \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 2\) instead of the correct answer \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 1\).

Errors while selecting the answer

1. Confusing rate with time in the final step

After correctly calculating that \(\mathrm{a} + \mathrm{b} + \mathrm{c} = 1\) tank per hour, students might mistakenly think this means it takes \(\frac{1}{1} = 1\) hour, but then second-guess themselves and select a different answer because they expect the combined rate to be faster than individual pairs. They may incorrectly choose a fraction like \(\frac{2}{3}\) or \(\frac{1}{2}\).

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for the tank capacity

Let's set the tank capacity to 12 units. This number works well because:

• 12 is divisible by the denominators in our given times (5, 2, and 1)
• It will give us clean integer rates for each pump combination

Step 2: Calculate combined rates for each pump pair

Using Rate = Work ÷ Time:

• Pumps A + B: \(\mathrm{Rate} = 12 \div \frac{6}{5} = 12 \times \frac{5}{6} = 10\) units per hour
• Pumps A + C: \(\mathrm{Rate} = 12 \div \frac{3}{2} = 12 \times \frac{2}{3} = 8\) units per hour
• Pumps B + C: \(\mathrm{Rate} = 12 \div 2 = 6\) units per hour

Step 3: Find the combined rate of all three pumps

We have the system:

• \(\mathrm{A} + \mathrm{B} = 10\)
• \(\mathrm{A} + \mathrm{C} = 8\)
• \(\mathrm{B} + \mathrm{C} = 6\)

Adding all three equations: \(2\mathrm{A} + 2\mathrm{B} + 2\mathrm{C} = 24\)

Therefore: \(\mathrm{A} + \mathrm{B} + \mathrm{C} = 12\) units per hour

Step 4: Calculate time for all three pumps working together

\(\mathrm{Time} = \mathrm{Work} \div \mathrm{Rate} = 12 \div 12 = 1\) hour

Answer: E (1 hour)