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Four banks offered home equity loans at the rates of \(4\frac{7}{8}\) percent, \(5\frac{1}{4}\) percent, \(4\frac{3}{5}\) percent, and \(5\frac{1}{8}\) percent, respectively. The greatest difference between any two of these rates, expressed as a decimal, is between
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:
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
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
For Bank 2: \(5\frac{1}{4}\) percent
For Bank 3: \(4\frac{3}{5}\) percent
For Bank 4: \(5\frac{1}{8}\) percent
Our four rates in decimal form are: \(4.875\%\), \(5.25\%\), \(4.6\%\), and \(5.125\%\)
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
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 \(0.0065\):
The greatest difference between any two rates is \(0.0065\), which falls in the range \(0.005\) and \(0.01\).
The answer is C.
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.
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.
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 \(0.6\).
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.
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.
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.