Four banks offered home equity loans at the rates of 47/8 percent, 51/4 percent, 43/5 percent, and 51/8 percent, respectively....

1. Translate the problem requirements: Convert four mixed number percentages to decimals, then find the greatest difference between any two rates and determine which range this difference falls into
2. Convert mixed numbers to decimal form: Transform each fractional percentage into its decimal equivalent for easier comparison
3. Identify maximum and minimum values: Determine the highest and lowest rates to find the greatest possible difference
4. Calculate the difference and match to ranges: Compute the difference between extreme values and identify which answer choice range contains this result

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we need to do. We have four banks offering home equity loans at different interest rates, and each rate is given as a mixed number percentage. Our job is to find the biggest difference between any two of these rates and see which range it falls into.

The four rates are:

• Bank 1: \(4\frac{7}{8}\) percent
• Bank 2: \(5\frac{1}{4}\) percent
• Bank 3: \(4\frac{3}{5}\) percent
• Bank 4: \(5\frac{1}{8}\) percent

We need to convert these to decimals, find the largest and smallest rates, calculate their difference, and match that difference to one of the given ranges.

Process Skill: TRANSLATE - Converting the problem statement into clear mathematical steps

2. Convert mixed numbers to decimal form

Now let's convert each mixed number to a decimal. Remember, a mixed number like \(4\frac{7}{8}\) means \(4 + \frac{7}{8}\).

For Bank 1: \(4\frac{7}{8}\) percent

• First convert the fraction: \(\frac{7}{8} = 7 \div 8 = 0.875\)
• So \(4\frac{7}{8} = 4 + 0.875 = 4.875\) percent

For Bank 2: \(5\frac{1}{4}\) percent

• First convert the fraction: \(\frac{1}{4} = 1 \div 4 = 0.25\)
• So \(5\frac{1}{4} = 5 + 0.25 = 5.25\) percent

For Bank 3: \(4\frac{3}{5}\) percent

• First convert the fraction: \(\frac{3}{5} = 3 \div 5 = 0.6\)
• So \(4\frac{3}{5} = 4 + 0.6 = 4.6\) percent

For Bank 4: \(5\frac{1}{8}\) percent

• First convert the fraction: \(\frac{1}{8} = 1 \div 8 = 0.125\)
• So \(5\frac{1}{8} = 5 + 0.125 = 5.125\) percent

Our four rates in decimal form are: \(4.875\%\), \(5.25\%\), \(4.6\%\), and \(5.125\%\)

3. Identify maximum and minimum values

To find the greatest difference, we need to identify the highest and lowest rates from our converted values.

Looking at our rates: \(4.875\%\), \(5.25\%\), \(4.6\%\), and \(5.125\%\)

The highest rate is \(5.25\%\) (Bank 2)
The lowest rate is \(4.6\%\) (Bank 3)

This makes sense because we want the biggest possible gap, which will always be between the extreme values.

Process Skill: SIMPLIFY - Recognizing that the maximum difference occurs between extreme values

4. Calculate the difference and match to ranges

Now we calculate the difference between the highest and lowest rates:

Greatest difference = \(5.25\% - 4.6\% = 0.65\%\)

But wait! The answer choices are given as regular decimals, not percentages. When we say \(0.65\%\), this means \(0.65\) per hundred, which as a regular decimal is:

\(0.65\% = 0.65 \div 100 = 0.0065\)

Now let's check which range contains \(0.0065\):

1. \(0.0001\) and \(0.0005\) → No, \(0.0065\) is larger
2. \(0.001\) and \(0.005\) → No, \(0.0065\) is larger than \(0.005\)
3. \(0.005\) and \(0.01\) → Yes! \(0.005 < 0.0065 < 0.01\) ✓
4. \(0.01\) and \(0.05\) → No, \(0.0065\) is smaller than \(0.01\)
5. \(0.05\) and \(0.1\) → No, \(0.0065\) is much smaller

Final Answer

The greatest difference between any two rates is \(0.0065\), which falls in the range \(0.005\) and \(0.01\).

The answer is C.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "greatest difference" means

Students might think they need to find all possible differences between pairs of rates rather than recognizing that the greatest difference will always be between the maximum and minimum values. This leads to unnecessary calculations and potential confusion.

2. Confusion about the final answer format

Students may not realize that the answer choices are expressed as regular decimals (like \(0.005\)) while the rates are given as percentages. They might plan to compare their percentage difference directly to the ranges without converting from percentage to decimal form.

Errors while executing the approach

1. Fraction conversion errors

Students commonly make mistakes when converting fractions to decimals, especially with fractions like \(\frac{7}{8}\) or \(\frac{3}{5}\). For example, they might incorrectly calculate \(\frac{7}{8}\) as \(0.78\) instead of \(0.875\), or \(\frac{3}{5}\) as \(0.35\) instead of \(0.6\).

2. Arithmetic errors in identifying extremes

After converting to decimals, students may incorrectly identify which rate is highest or lowest. For instance, they might think \(4.875\%\) is larger than \(5.125\%\) by focusing only on the decimal portion (\(0.875 > 0.125\)) while ignoring the whole number part.

3. Percentage to decimal conversion mistakes

When converting the final percentage difference (\(0.65\%\)) to a regular decimal, students often forget to divide by \(100\), giving \(0.65\) instead of \(0.0065\). This is a critical error that leads to selecting a completely wrong answer range.

Errors while selecting the answer

1. Boundary confusion in ranges

Even with the correct decimal value of \(0.0065\), students might incorrectly determine which range it falls into. They may select option B (\(0.001\) to \(0.005\)) thinking \(0.0065\) is "close enough" to \(0.005\), not realizing that \(0.0065 > 0.005\) means it doesn't fit in that range.