If K is the sum of the reciprocals of the consecutive integers from 43 to 48, inclusive, then K is...

1. Translate the problem requirements: K equals the sum \(\frac{1}{43} + \frac{1}{44} + \frac{1}{45} + \frac{1}{46} + \frac{1}{47} + \frac{1}{48}\), and we need to find which answer choice this sum is closest to
2. Analyze answer choice magnitudes for approximation strategy: Since answer choices range from \(\frac{1}{12} ≈ 0.083\) to \(\frac{1}{4} = 0.25\), we can use reasonable approximations rather than exact decimal calculations
3. Estimate the sum using representative values: Since all terms are close to \(\frac{1}{45}\), we can approximate the sum as \(6 \times \frac{1}{45}\) and compare to answer choices
4. Refine the estimate by considering the range: Check if our approximation falls within the correct answer choice range by noting that terms vary from \(\frac{1}{43}\) to \(\frac{1}{48}\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what we need to find. We're told that K is the sum of the reciprocals of consecutive integers from 43 to 48, inclusive.

In everyday language, a reciprocal of a number is just "1 divided by that number." So the reciprocal of 43 is \(\frac{1}{43}\), the reciprocal of 44 is \(\frac{1}{44}\), and so on.

Since we want consecutive integers from 43 to 48 inclusive, we need to add up:

\(\mathrm{K} = \frac{1}{43} + \frac{1}{44} + \frac{1}{45} + \frac{1}{46} + \frac{1}{47} + \frac{1}{48}\)

We have exactly 6 terms to add together, and we need to find which answer choice this sum is closest to.

Process Skill: TRANSLATE - Converting the phrase "sum of reciprocals" into the actual mathematical expression we need to evaluate.

2. Analyze answer choice magnitudes for approximation strategy

Before we start calculating, let's look at what our answer choices actually mean in decimal form. This will help us know how precise our approximation needs to be.

1. \(\frac{1}{12} ≈ 0.083\)
2. \(\frac{1}{10} = 0.10\)
3. \(\frac{1}{8} = 0.125\)
4. \(\frac{1}{6} ≈ 0.167\)
5. \(\frac{1}{4} = 0.25\)

Notice that these answer choices are reasonably spaced apart. The smallest gap is between \(\frac{1}{12}\) and \(\frac{1}{10}\), which is about 0.017. This means we don't need to calculate K to extreme precision - a reasonable approximation should clearly point us to the right answer.

3. Estimate the sum using representative values

Here's the key insight: all our terms (\(\frac{1}{43}, \frac{1}{44}, \frac{1}{45}, \frac{1}{46}, \frac{1}{47}, \frac{1}{48}\)) are pretty close to each other in value. They're all around \(\frac{1}{45}\), which is roughly in the middle.

So instead of calculating each term exactly, let's approximate: if we pretend each term equals \(\frac{1}{45}\), then our sum would be:

\(\mathrm{K} ≈ 6 \times \frac{1}{45} = \frac{6}{45} = \frac{2}{15}\)

Now let's convert \(\frac{2}{15}\) to decimal form: \(\frac{2}{15} ≈ 0.133\)

Looking at our answer choices from step 2, this value of 0.133 is very close to choice C: \(\frac{1}{8} = 0.125\)

4. Refine the estimate by considering the range

Let's double-check our approximation by considering that our terms aren't exactly equal to \(\frac{1}{45}\).

Our actual terms range from \(\frac{1}{43}\) (the largest) to \(\frac{1}{48}\) (the smallest).

The largest possible sum would be: \(6 \times \frac{1}{43} = \frac{6}{43} ≈ 0.140\)

The smallest possible sum would be: \(6 \times \frac{1}{48} = \frac{6}{48} = \frac{1}{8} = 0.125\)

So our actual sum K must be between 0.125 and 0.140.

From our answer choices, only \(\frac{1}{8} = 0.125\) falls in this range. The next closest choice is \(\frac{1}{6} ≈ 0.167\), which is too large.

This confirms that K is closest to \(\frac{1}{8}\).

Process Skill: APPLY CONSTRAINTS - Using the range of our terms to verify our approximation is reasonable.

5. Final Answer

K is closest in value to C. \(\frac{1}{8}\)

Our approximation using \(6 \times \frac{1}{45} ≈ 0.133\) pointed to this answer, and our range check confirmed that the actual value of K must be between 0.125 and 0.140, making \(\frac{1}{8} = 0.125\) the closest answer choice.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "consecutive integers from 43 to 48, inclusive"
Students might incorrectly count the number of terms. They may think there are 5 terms (48-43=5) instead of 6 terms, forgetting that "inclusive" means both 43 and 48 are included. This leads to calculating \(\mathrm{K} = \frac{1}{43} + \frac{1}{44} + \frac{1}{45} + \frac{1}{46} + \frac{1}{47}\), missing the \(\frac{1}{48}\) term.

2. Attempting exact calculation instead of strategic approximation
Students may try to find a common denominator for all fractions (\(\frac{1}{43} + \frac{1}{44} + \frac{1}{45} + \frac{1}{46} + \frac{1}{47} + \frac{1}{48}\)) and perform exact arithmetic. This approach is extremely time-consuming and error-prone on the GMAT, where approximation strategies are more effective.

3. Poor choice of representative value for approximation
Students might choose an inappropriate representative value, such as using \(\frac{1}{40}\) or \(\frac{1}{50}\) instead of \(\frac{1}{45}\) (which is roughly in the middle of our range). Using \(\frac{1}{40}\) would give \(6 \times \frac{1}{40} = 0.15\), while using \(\frac{1}{50}\) gives \(6 \times \frac{1}{50} = 0.12\), both leading to less accurate approximations.

Errors while executing the approach

1. Arithmetic errors in approximation calculations
When calculating \(6 \times \frac{1}{45} = \frac{6}{45} = \frac{2}{15}\), students may make errors in simplification or decimal conversion. For example, incorrectly simplifying \(\frac{6}{45}\) to \(\frac{1}{8}\) directly, or miscalculating \(\frac{2}{15}\) as 0.113 instead of approximately 0.133.

2. Incorrect decimal conversions of answer choices
Students may convert the answer choices incorrectly, such as calculating \(\frac{1}{8} = 0.115\) instead of 0.125, or \(\frac{1}{6} = 0.15\) instead of approximately 0.167. These errors make it difficult to compare their calculated value with the correct answer choices.

Errors while selecting the answer

1. Selecting based on insufficient comparison
After calculating K ≈ 0.133, students might hastily select \(\frac{1}{6} ≈ 0.167\) because they see it's "close" without carefully comparing it to \(\frac{1}{8} = 0.125\). They fail to recognize that 0.133 is actually much closer to 0.125 than to 0.167 (difference of 0.008 vs 0.034).