The harmonic mean of n numbers is the reciprocal of the arithmetic mean of their reciprocals. What is the harmonic...

1. Translate the problem requirements: The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals. This means: (1) find the reciprocal of each number, (2) calculate the arithmetic mean of those reciprocals, (3) take the reciprocal of that result.
2. Calculate the reciprocals of given numbers: Convert each of the four numbers (2, 4, 6, 12) into their reciprocal form to prepare for the arithmetic mean calculation.
3. Find the arithmetic mean of the reciprocals: Add all the reciprocals together and divide by the count of numbers to get the average of the reciprocals.
4. Take the reciprocal of the arithmetic mean: The final step applies the harmonic mean definition by taking the reciprocal of the result from the previous step.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what "harmonic mean" means in plain English. The problem tells us that the harmonic mean is "the reciprocal of the arithmetic mean of their reciprocals." This sounds complicated, but let's think of it step by step:

• First, we need to find the "reciprocal" of each number (flip it upside down: 2 becomes \(\frac{1}{2}\))
• Then, we find the "arithmetic mean" of those flipped numbers (just the regular average)
• Finally, we take the "reciprocal" of that average (flip it upside down again)

So our process is: Numbers → Reciprocals → Average of reciprocals → Flip that average

Our numbers are: 2, 4, 6, and 12

Process Skill: TRANSLATE - Converting the definition into a clear step-by-step process

2. Calculate the reciprocals of given numbers

Now let's flip each number upside down to get its reciprocal:

• Reciprocal of 2 = \(\frac{1}{2}\)
• Reciprocal of 4 = \(\frac{1}{4}\)
• Reciprocal of 6 = \(\frac{1}{6}\)
• Reciprocal of 12 = \(\frac{1}{12}\)

So our list of reciprocals is: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{6}\), \(\frac{1}{12}\)

3. Find the arithmetic mean of the reciprocals

Now we need to find the average of these four fractions: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{6}\), \(\frac{1}{12}\)

To add fractions, we need a common denominator. Let's find the least common multiple of 2, 4, 6, and 12:
• 12 is divisible by 2, 4, 6, and 12, so our common denominator is 12

Converting each fraction:
• \(\frac{1}{2} = \frac{6}{12}\)
• \(\frac{1}{4} = \frac{3}{12}\)
• \(\frac{1}{6} = \frac{2}{12}\)
• \(\frac{1}{12} = \frac{1}{12}\)

Adding them up: \(\frac{6}{12} + \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{12}{12} = 1\)

Since we have 4 numbers, the arithmetic mean is: \(1 ÷ 4 = \frac{1}{4}\)

Process Skill: SIMPLIFY - Using common denominators to make fraction arithmetic manageable

4. Take the reciprocal of the arithmetic mean

The final step is to flip our arithmetic mean upside down:

The arithmetic mean of the reciprocals was \(\frac{1}{4}\)
The reciprocal of \(\frac{1}{4}\) is \(\frac{4}{1} = 4\)

Therefore, the harmonic mean of 2, 4, 6, and 12 is 4.

Final Answer

The harmonic mean is 4, which corresponds to answer choice D.

We can verify this makes sense: the harmonic mean (4) falls between the smallest (2) and largest (12) of our original numbers, which is what we'd expect for any type of average.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the definition of harmonic mean
Students often struggle with the complex definition "reciprocal of the arithmetic mean of their reciprocals" and may confuse it with other types of means (arithmetic or geometric). They might try to directly average the numbers 2, 4, 6, and 12 instead of following the three-step process: find reciprocals → find arithmetic mean → take reciprocal again.

2. Getting lost in the multi-step process
The harmonic mean requires three distinct operations in sequence. Students may understand each individual step but lose track of the overall process, potentially skipping steps or performing them in the wrong order.

Errors while executing the approach

1. Making errors when finding common denominators
When adding the fractions \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{6}\), and \(\frac{1}{12}\), students often struggle to find the correct least common multiple (12) or make conversion errors. For example, they might incorrectly convert \(\frac{1}{2}\) to \(\frac{3}{12}\) instead of \(\frac{6}{12}\), or miscalculate when adding the fractions together.

2. Arithmetic errors in the division step
After correctly finding that the sum of reciprocals equals 1, students may forget to divide by the number of terms (4) to get the arithmetic mean, or they may incorrectly calculate \(1 ÷ 4\), leading to wrong intermediate results.

3. Confusion with taking the final reciprocal
Students may correctly find that the arithmetic mean of reciprocals is \(\frac{1}{4}\) but then forget the final step of taking its reciprocal, or they may incorrectly calculate the reciprocal of \(\frac{1}{4}\) as something other than 4.

Errors while selecting the answer

1. Selecting an intermediate result instead of the final answer
Students might select \(\frac{1}{4}\) (the arithmetic mean of the reciprocals) instead of 4 (the harmonic mean), confusing an intermediate step with the final result. This could lead them to choose answer choice B instead of the correct answer D.