Which of the following fractions is closest to 1/2?

1. Translate the problem requirements: We need to find which fraction has the smallest distance from \(\frac{1}{2}\), meaning we're looking for the fraction that is closest in value to 0.5
2. Convert the comparison to a common reference point: Since comparing to \(\frac{1}{2}\) directly is easiest, we'll determine whether each fraction is greater than or less than \(\frac{1}{2}\) and by how much
3. Use the halfway test for quick comparison: For any fraction \(\frac{\mathrm{a}}{\mathrm{b}}\), if \(2\mathrm{a} = \mathrm{b}\), then the fraction equals \(\frac{1}{2}\). If \(2\mathrm{a} > \mathrm{b}\), it's greater than \(\frac{1}{2}\). If \(2\mathrm{a} < \mathrm{b}\), it's less than \(\frac{1}{2}\)
4. Calculate the distance from \(\frac{1}{2}\) for the closest candidates: Focus on the fractions that appear closest based on the halfway test and find their exact distances from \(\frac{1}{2}\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're looking for. We want to find which fraction is closest to \(\frac{1}{2}\). This means we need to find which fraction has the smallest distance from \(\frac{1}{2}\).

Think of it like this: if \(\frac{1}{2}\) is our target on a number line, we want to find which of our answer choices lands nearest to that target. The "distance" is how far away each fraction is from \(\frac{1}{2}\), whether it's bigger or smaller than \(\frac{1}{2}\).

2. Convert the comparison to a common reference point

Since we're comparing everything to \(\frac{1}{2}\), let's make \(\frac{1}{2}\) our reference point. For each fraction, we'll figure out:

• Is it bigger than \(\frac{1}{2}\) or smaller than \(\frac{1}{2}\)?
• By how much is it bigger or smaller?

This way, we can easily see which one is closest to our target of \(\frac{1}{2}\).

3. Use the halfway test for quick comparison

Here's a simple trick to quickly tell if a fraction is bigger or smaller than \(\frac{1}{2}\):

The Halfway Test: For any fraction like \(\frac{\mathrm{a}}{\mathrm{b}}\), double the top number (numerator) and compare it to the bottom number (denominator):

• If \(2 \times \text{(top number)} = \text{bottom number}\) → the fraction equals \(\frac{1}{2}\)
• If \(2 \times \text{(top number)} > \text{bottom number}\) → the fraction is greater than \(\frac{1}{2}\)
• If \(2 \times \text{(top number)} < \text{bottom number}\) → the fraction is less than \(\frac{1}{2}\)

Let's apply this test to each answer choice:

1. \(\frac{4}{7}\): \(2 \times 4 = 8\), and \(8 > 7\), so \(\frac{4}{7} > \frac{1}{2}\)
2. \(\frac{5}{9}\): \(2 \times 5 = 10\), and \(10 > 9\), so \(\frac{5}{9} > \frac{1}{2}\)
3. \(\frac{6}{11}\): \(2 \times 6 = 12\), and \(12 > 11\), so \(\frac{6}{11} > \frac{1}{2}\)
4. \(\frac{7}{13}\): \(2 \times 7 = 14\), and \(14 > 13\), so \(\frac{7}{13} > \frac{1}{2}\)
5. \(\frac{9}{16}\): \(2 \times 9 = 18\), and \(18 > 16\), so \(\frac{9}{16} > \frac{1}{2}\)

Interesting! All fractions are greater than \(\frac{1}{2}\). Now we need to find which one is closest to \(\frac{1}{2}\) (meaning the smallest amount greater than \(\frac{1}{2}\)).

4. Calculate the distance from \(\frac{1}{2}\) for the closest candidates

Since all fractions are greater than \(\frac{1}{2}\), we need to find how much greater each one is. We'll calculate the exact distance by subtracting \(\frac{1}{2}\) from each fraction.

Distance = Fraction - \(\frac{1}{2}\)

Let's calculate for each option:

A) \(\frac{4}{7} - \frac{1}{2}\):

To subtract fractions, we need a common denominator. The common denominator of 7 and 2 is 14.

\(\frac{4}{7} = \frac{8}{14}\) and \(\frac{1}{2} = \frac{7}{14}\)

Distance = \(\frac{8}{14} - \frac{7}{14} = \frac{1}{14}\)

B) \(\frac{5}{9} - \frac{1}{2}\):

Common denominator of 9 and 2 is 18.

\(\frac{5}{9} = \frac{10}{18}\) and \(\frac{1}{2} = \frac{9}{18}\)

Distance = \(\frac{10}{18} - \frac{9}{18} = \frac{1}{18}\)

C) \(\frac{6}{11} - \frac{1}{2}\):

Common denominator of 11 and 2 is 22.

\(\frac{6}{11} = \frac{12}{22}\) and \(\frac{1}{2} = \frac{11}{22}\)

Distance = \(\frac{12}{22} - \frac{11}{22} = \frac{1}{22}\)

D) \(\frac{7}{13} - \frac{1}{2}\):

Common denominator of 13 and 2 is 26.

\(\frac{7}{13} = \frac{14}{26}\) and \(\frac{1}{2} = \frac{13}{26}\)

Distance = \(\frac{14}{26} - \frac{13}{26} = \frac{1}{26}\)

E) \(\frac{9}{16} - \frac{1}{2}\):

Common denominator of 16 and 2 is 16.

\(\frac{9}{16} = \frac{9}{16}\) and \(\frac{1}{2} = \frac{8}{16}\)

Distance = \(\frac{9}{16} - \frac{8}{16} = \frac{1}{16}\)

Now we compare the distances: \(\frac{1}{14}\), \(\frac{1}{18}\), \(\frac{1}{22}\), \(\frac{1}{26}\), and \(\frac{1}{16}\)

Since all numerators are 1, the fraction with the largest denominator will be the smallest fraction.

The largest denominator is 26, so \(\frac{1}{26}\) is the smallest distance.

5. Final Answer

The fraction closest to \(\frac{1}{2}\) is \(\frac{7}{13}\), which corresponds to answer choice D.

This makes sense because \(\frac{7}{13}\) has the smallest distance from \(\frac{1}{2}\) (only \(\frac{1}{26}\) away), making it the closest to our target value.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "closest" to mean exact equality: Students might look for a fraction that equals exactly \(\frac{1}{2}\) instead of understanding that "closest" means finding the smallest distance from \(\frac{1}{2}\). This leads them to waste time trying to find an exact match rather than comparing distances.

2. Choosing an inefficient comparison method: Instead of using the strategic halfway test (comparing \(2 \times \text{numerator}\) to denominator), students might immediately convert all fractions to decimals or find common denominators for all fractions at once. This makes the problem much more time-consuming and error-prone.

3. Not recognizing the distance concept: Students might try to rank the fractions from smallest to largest without understanding that they need to measure how far each fraction is from the specific target of \(\frac{1}{2}\), leading to incorrect comparison strategies.

Errors while executing the approach

1. Arithmetic errors in the halfway test: When applying the test "\(2 \times \text{numerator}\) compared to denominator," students commonly make simple multiplication errors. For example, calculating \(2 \times 7 = 15\) instead of 14, or \(2 \times 6 = 11\) instead of 12, which leads to incorrect conclusions about whether fractions are greater or less than \(\frac{1}{2}\).

2. Common denominator calculation mistakes: When finding distances like \(\left(\frac{7}{13} - \frac{1}{2}\right)\), students often make errors in finding common denominators or in the conversion process. For instance, incorrectly converting \(\frac{7}{13}\) to \(\frac{13}{26}\) instead of \(\frac{14}{26}\), or converting \(\frac{1}{2}\) to \(\frac{12}{26}\) instead of \(\frac{13}{26}\).

3. Sign errors in distance calculation: Students might subtract in the wrong direction (calculating \(\frac{1}{2} - \frac{7}{13}\) instead of \(\frac{7}{13} - \frac{1}{2}\)) or forget that distance is always positive, leading them to work with negative values that complicate their comparisons.

Errors while selecting the answer

1. Confusing smallest distance with largest denominator: After finding that all distances have the form \(\frac{1}{\mathrm{n}}\), students might incorrectly choose the fraction with the smallest denominator (\(\frac{1}{14}\)) thinking it's the smallest distance, rather than recognizing that \(\frac{1}{26}\) is actually the smallest because larger denominators create smaller fractions.

2. Selecting based on the original fraction size rather than distance: Students might choose the largest original fraction (like \(\frac{9}{16}\)) thinking that "closest to \(\frac{1}{2}\)" means "largest value," rather than understanding they need the fraction with the smallest distance from \(\frac{1}{2}\).