A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value...

1. Translate the problem requirements: Convert the repeating decimal \(0.00\overline{12}\) to a fraction, then calculate \((10^4 - 10^2)\) times this value
2. Convert repeating decimal to fraction: Use algebraic manipulation to express \(0.00\overline{12}\) as a simple fraction
3. Simplify the coefficient: Calculate \(10^4 - 10^2 = 10,000 - 100 = 9,900\)
4. Multiply and simplify: Multiply 9,900 by the fraction and reduce to find the final answer

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with here. We have a repeating decimal \(0.00\overline{12}\), which means \(0.001212121212...\) where the digits "12" repeat forever after the first two zeros.

We need to find the value of \((10^4 - 10^2) \times 0.00\overline{12}\).

To solve this completely, we'll need to convert that repeating decimal into a fraction so we can work with it more easily.

Process Skill: TRANSLATE - Converting the bar notation into mathematical understanding

2. Convert repeating decimal to fraction

Let's call our repeating decimal x, so \(\mathrm{x} = 0.00\overline{12} = 0.001212121212...\)

Here's the key insight: if we multiply this by the right power of 10, we can line up the repeating parts and subtract to eliminate the repetition.

Since our repeating block "12" has 2 digits and starts after 2 decimal places:

- Multiply by \(10^2 = 100\): \(100\mathrm{x} = 0.121212121212...\)
- Multiply by \(10^4 = 10,000\): \(10,000\mathrm{x} = 12.121212121212...\)

Now subtract the first equation from the second:
\(10,000\mathrm{x} - 100\mathrm{x} = 12.121212... - 0.121212...\)
\(9,900\mathrm{x} = 12\)

Therefore: \(\mathrm{x} = \frac{12}{9,900}\)

We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 12:
\(\mathrm{x} = \frac{12}{9,900} = \frac{1}{825}\)

So \(0.00\overline{12} = \frac{1}{825}\)

3. Simplify the coefficient

Now let's calculate the coefficient \((10^4 - 10^2)\):
\(10^4 - 10^2 = 10,000 - 100 = 9,900\)

Notice something interesting here - this is exactly the same number we got in the denominator when we were finding our fraction!

4. Multiply and simplify

Now we can calculate our final answer:
\((10^4 - 10^2) \times 0.00\overline{12} = 9,900 \times \left(\frac{1}{825}\right)\)

But wait - let's use our original fraction before we simplified it:
\(9,900 \times \left(\frac{12}{9,900}\right) = \frac{9,900 \times 12}{9,900} = 12\)

The 9,900 terms cancel out completely, leaving us with simply 12.

Process Skill: SIMPLIFY - Recognizing the algebraic cancellation that makes this problem much simpler than it initially appears

5. Final Answer

The value of \((10^4 - 10^2)(0.00\overline{12}) = 12\)

Checking against our answer choices, this matches choice (E) 12.

This elegant result shows that the arithmetic was designed to create this clean cancellation - a common feature in GMAT problems where complex-looking expressions often simplify beautifully.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the repeating decimal notation
Students often misinterpret \(0.00\overline{12}\) as 0.0012 (a terminating decimal) instead of recognizing it as \(0.001212121212...\) where "12" repeats infinitely. This fundamental misunderstanding leads to completely incorrect calculations throughout the problem.

Faltering Point 2: Choosing an overly complex approach
Students might attempt to work directly with the repeating decimal in its decimal form rather than converting it to a fraction first. This makes the multiplication extremely difficult and error-prone, when converting to a fraction creates a much cleaner solution path.

Faltering Point 3: Incorrect setup for converting repeating decimal to fraction
Students often struggle with determining the correct powers of 10 to use when converting repeating decimals. Since \(0.00\overline{12}\) has the repeating block starting after 2 decimal places and the block has 2 digits, students might incorrectly use \(10^1\) and \(10^3\) instead of the correct \(10^2\) and \(10^4\).

Errors while executing the approach

Faltering Point 1: Arithmetic errors in the subtraction step
When subtracting \(100\mathrm{x} = 0.121212...\) from \(10,000\mathrm{x} = 12.121212...\), students often make errors in aligning the decimal places or in the subtraction itself, leading to incorrect values like \(9,900\mathrm{x} = 11\) instead of \(9,900\mathrm{x} = 12\).

Faltering Point 2: Incorrect simplification of the fraction
Students might incorrectly simplify \(\frac{12}{9,900}\) by dividing by the wrong common factor, or they might make errors in finding the GCD. Some might also prematurely simplify before recognizing the elegant cancellation that occurs in the final multiplication.

Faltering Point 3: Calculation errors with powers of 10
Students frequently make basic arithmetic errors when calculating \(10^4 - 10^2\), either miscalculating the individual powers (writing \(10^4 = 1,000\) instead of 10,000) or making subtraction errors (getting 9,000 instead of 9,900).

Errors while selecting the answer

Faltering Point 1: Selecting a decimal answer when the result is a whole number
Even if students calculate correctly and arrive at 12, they might doubt their answer because it seems "too simple" for such a complex-looking problem. This could lead them to select answer choice (C) 1.2, thinking they made a decimal place error, or (B) \(0.\overline{12}\), thinking the answer should still be a repeating decimal.