Which of the following fractions has the greatest value?

1. Translate the problem requirements: We need to compare five fractions and determine which has the greatest value. Each fraction has powers of 2 and 5 in the denominator, so we're looking for the fraction that represents the largest decimal value.
2. Establish a common comparison framework: Since all denominators contain powers of 2 and 5, we can rewrite each fraction with a common denominator structure to make direct comparison possible.
3. Convert to decimal equivalents for direct comparison: Calculate the actual decimal values of each fraction to determine which is largest, focusing on efficient calculation methods.
4. Verify the largest value: Confirm our answer by double-checking the calculation of the winning fraction and ensuring it's indeed larger than the others.

Execution of Strategic Approach

1. Translate the problem requirements

We need to find which of these five fractions gives us the biggest number when we calculate it out. Think of it like this: if you had to choose between getting \(\frac{1}{4}\) of a pizza versus \(\frac{1}{2}\) of a pizza, you'd want \(\frac{1}{2}\) because it's bigger. Here we have more complex fractions, but the same principle applies.

Each fraction has a numerator (top number) and a denominator (bottom number) that contains powers of 2 and powers of 5. The bigger the numerator and the smaller the denominator, the larger our fraction will be.

Let's write out what we're comparing:

• Option A: \(6\) divided by \((2^2 \times 5^2)\)
• Option B: \(1\) divided by \((2^3 \times 5^2)\)
• Option C: \(28\) divided by \((2^2 \times 5^3)\)
• Option D: \(62\) divided by \((2^3 \times 5^3)\)
• Option E: \(122\) divided by \((2^4 \times 5^3)\)

Process Skill: TRANSLATE - Converting the mathematical notation into plain language understanding

2. Establish a common comparison framework

To compare these fractions fairly, let's first calculate what each denominator actually equals. This is like finding a common way to measure all our pizza slices.

Let's work out the denominators step by step:

• \(2^2 = 4\) and \(5^2 = 25\), so \(2^2 \times 5^2 = 4 \times 25 = 100\)
• \(2^3 = 8\) and \(5^2 = 25\), so \(2^3 \times 5^2 = 8 \times 25 = 200\)
• \(2^2 = 4\) and \(5^3 = 125\), so \(2^2 \times 5^3 = 4 \times 125 = 500\)
• \(2^3 = 8\) and \(5^3 = 125\), so \(2^3 \times 5^3 = 8 \times 125 = 1000\)
• \(2^4 = 16\) and \(5^3 = 125\), so \(2^4 \times 5^3 = 16 \times 125 = 2000\)

Now our fractions look much cleaner:

• Option A: \(\frac{6}{100}\)
• Option B: \(\frac{1}{200}\)
• Option C: \(\frac{28}{500}\)
• Option D: \(\frac{62}{1000}\)
• Option E: \(\frac{122}{2000}\)

Process Skill: SIMPLIFY - Breaking down complex expressions into manageable numbers

3. Convert to decimal equivalents for direct comparison

Now let's convert each fraction to a decimal so we can easily see which is biggest. This is like converting all our measurements to the same units.

• Option A: \(\frac{6}{100} = 0.06\)
• Option B: \(\frac{1}{200} = 0.005\)
• Option C: \(\frac{28}{500} = 0.056\)
• Option D: \(\frac{62}{1000} = 0.062\)
• Option E: \(\frac{122}{2000} = 0.061\)

Looking at these decimal values:

• \(0.06\) (Option A)
• \(0.005\) (Option B) - clearly the smallest
• \(0.056\) (Option C)
• \(0.062\) (Option D) - this looks like the largest
• \(0.061\) (Option E)

Option D gives us \(0.062\), which is larger than all the others.

4. Verify the largest value

Let's double-check our calculation for Option D and compare it with the closest competitor (Option E):

Option D: \(\frac{62}{1000} = 0.062\)
Option E: \(\frac{122}{2000} = \frac{122}{2000} = \frac{61}{1000} = 0.061\)

Indeed, \(0.062 > 0.061\), confirming that Option D is larger.

We can also verify this makes sense: Option D has a numerator of \(62\) and when we scale Option E to the same denominator \((1000)\), it becomes \(\frac{61}{1000}\). Since \(62 > 61\), Option D wins.

Final Answer

The fraction with the greatest value is Option D: \(\frac{62}{(2^3)(5^3)}\), which equals \(0.062\). This fraction is larger than all other options when converted to decimal form.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the comparison task: Students may think they need to find a common denominator for all fractions (like adding fractions) rather than simply determining which has the greatest value. This leads to unnecessarily complex calculations and potential errors.

2. Attempting to use cross-multiplication incorrectly: When comparing multiple fractions, students might try to use cross-multiplication methods meant for comparing just two fractions, leading to confusion when dealing with five different options simultaneously.

3. Overlooking the decimal conversion approach: Students may get intimidated by the exponential notation and attempt complex algebraic manipulations instead of recognizing that converting to decimals provides the most straightforward comparison method.

Errors while executing the approach

1. Calculation errors in computing powers: Students frequently make mistakes when calculating exponentials, especially confusing \(2^3 = 8\) vs \(2^4 = 16\), or \(5^2 = 25\) vs \(5^3 = 125\). These errors cascade through the entire solution.

2. Arithmetic mistakes in multiplication: When computing denominators like \(2^3 \times 5^3 = 8 \times 125\), students often make multiplication errors (getting \(1125\) instead of \(1000\), for example), leading to incorrect decimal conversions.

3. Decimal conversion errors: Students may incorrectly convert fractions to decimals, such as computing \(\frac{28}{500}\) as \(0.56\) instead of \(0.056\), often dropping or misplacing decimal places.

Errors while selecting the answer

1. Misreading decimal comparisons: Students may incorrectly order decimals like \(0.061\) and \(0.062\), thinking \(0.061\) is larger because \(61 > 62\), forgetting to consider the decimal place values properly.

2. Selecting based on numerator size alone: Seeing that option E has the largest numerator \((122)\), students might incorrectly assume it must be the largest fraction without considering the denominator's impact.