If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the...

1. Translate the problem requirements: We need to find \(\mathrm{a} - \mathrm{b}\) where \(\mathrm{a}\) is the sum of specific fractions (\(1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}\)) and \(\mathrm{b}\) is defined as \(1 + \frac{1}{4}\mathrm{a}\). The question asks for the numerical difference between these two expressions.
2. Calculate the value of 'a' directly: Add the given fractions by finding a common denominator and computing the exact sum.
3. Express 'b' in terms of the calculated value of 'a': Substitute the value of \(\mathrm{a}\) into the expression \(\mathrm{b} = 1 + \frac{1}{4}\mathrm{a}\) to get the numerical value of \(\mathrm{b}\).
4. Compute the final difference: Calculate \(\mathrm{a} - \mathrm{b}\) using the computed values and simplify to match one of the answer choices.

Execution of Strategic Approach

1. Translate the problem requirements

Let's understand what we're being asked to find. We have two values:

- '\(\mathrm{a}\)' is the sum of four specific fractions: \(1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}\)

- '\(\mathrm{b}\)' is defined as \(1 + \frac{1}{4}\) times the value of '\(\mathrm{a}\)'

Our goal is to find the difference \(\mathrm{a} - \mathrm{b}\). This means we need to calculate both '\(\mathrm{a}\)' and '\(\mathrm{b}\)' first, then subtract '\(\mathrm{b}\)' from '\(\mathrm{a}\)'.

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

2. Calculate the value of 'a' directly

Now let's find the exact value of \(\mathrm{a}\) by adding up all the fractions. To add fractions, we need a common denominator.

Looking at our fractions: \(1\), \(\frac{1}{4}\), \(\frac{1}{16}\), and \(\frac{1}{64}\), we can see that:

- \(1 = \frac{64}{64}\)

- \(\frac{1}{4} = \frac{16}{64}\)

- \(\frac{1}{16} = \frac{4}{64}\)

- \(\frac{1}{64} = \frac{1}{64}\)

So: \(\mathrm{a} = \frac{64}{64} + \frac{16}{64} + \frac{4}{64} + \frac{1}{64}\)

Adding the numerators: \(\mathrm{a} = \frac{64 + 16 + 4 + 1}{64} = \frac{85}{64}\)

3. Express 'b' in terms of the calculated value of 'a'

Now we can find '\(\mathrm{b}\)' using the formula \(\mathrm{b} = 1 + \frac{1}{4}\mathrm{a}\).

We know \(\mathrm{a} = \frac{85}{64}\), so:

\(\mathrm{b} = 1 + \frac{1}{4} \cdot \frac{85}{64}\)

\(\mathrm{b} = 1 + \frac{85}{256}\)

To add these, we need a common denominator:

\(\mathrm{b} = \frac{256}{256} + \frac{85}{256} = \frac{256 + 85}{256} = \frac{341}{256}\)

4. Compute the final difference

Now we can calculate \(\mathrm{a} - \mathrm{b}\):

\(\mathrm{a} - \mathrm{b} = \frac{85}{64} - \frac{341}{256}\)

To subtract these fractions, we need a common denominator. Since \(256 = 4 \times 64\), we can write:

\(\frac{85}{64} = \frac{340}{256}\)

Therefore:

\(\mathrm{a} - \mathrm{b} = \frac{340}{256} - \frac{341}{256} = \frac{340 - 341}{256} = -\frac{1}{256}\)

Final Answer

The value of \(\mathrm{a} - \mathrm{b} = -\frac{1}{256}\), which matches answer choice B.

Let's verify: We found \(\mathrm{a} = \frac{85}{64}\) and \(\mathrm{b} = \frac{341}{256}\). Converting \(\mathrm{a}\) to the same denominator: \(\mathrm{a} = \frac{340}{256}\). So \(\mathrm{a} - \mathrm{b} = \frac{340}{256} - \frac{341}{256} = -\frac{1}{256}\). ✓

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the definition of variable b

Students may misread the expression \(\mathrm{b} = 1 + \frac{1}{4}\mathrm{a}\) as \(\mathrm{b} = \left(1 + \frac{1}{4}\right)\mathrm{a}\), which would mean \(\mathrm{b} = \frac{5}{4}\mathrm{a}\). This changes the entire calculation and leads to a completely different answer. The parentheses and order of operations are crucial here.

2. Attempting to use geometric series formula incorrectly

Some students might recognize that \(\mathrm{a}\) looks like it could be part of a geometric series (with first term 1 and ratio \(\frac{1}{4}\)), but may try to apply the infinite series formula when we only have 4 specific terms. This can lead to using \(\mathrm{a} = \frac{1}{1-\frac{1}{4}} = \frac{4}{3}\) instead of calculating the actual sum.

Errors while executing the approach

1. Common denominator errors when calculating 'a'

When converting fractions to have denominator 64, students often make mistakes such as writing \(\frac{1}{4} = \frac{4}{64}\) instead of \(\frac{16}{64}\), or \(\frac{1}{16} = \frac{1}{64}\) instead of \(\frac{4}{64}\). These errors in fraction conversion lead to an incorrect value of \(\mathrm{a}\).

2. Arithmetic mistakes when computing (1/4) × (85/64)

Students may incorrectly calculate \(\frac{1}{4} \times \frac{85}{64}\) as \(\frac{85}{16}\) instead of \(\frac{85}{256}\), forgetting that they need to multiply both the numerator by numerator and denominator by denominator: \(\frac{1 \times 85}{4 \times 64} = \frac{85}{256}\).

3. Converting to common denominators for subtraction

When computing \(\mathrm{a} - \mathrm{b}\), students need to express both fractions with the same denominator. They might incorrectly convert \(\frac{85}{64}\) to \(\frac{85}{256}\) instead of \(\frac{340}{256}\), or make errors when performing \(\frac{340}{256} - \frac{341}{256}\).

Errors while selecting the answer

No likely faltering points - the final calculation directly gives \(-\frac{1}{256}\), which matches exactly with answer choice B, leaving little room for selection errors.