"What Is the Value of (10⁴ − 10²) × 0.0012̄?" — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value of $(10^4 - 10^2) \times 0.00\overline{12}$?

(A) 0

(B) $0.\overline{12}$

(C) 1.2

(D) 10

(E) 12

Try it before reading on.

The Critical Trap: Exponent Shortcuts on Subtraction

About 25% of test takers miss this problem, and nearly half of those misses come from one specific error: simplifying $10^4 - 10^2$ as $10^2$.

This feels right. The numbers are close, the exponents look like they should subtract, and the shortcut rule "same base, subtract exponents" is a real rule. But it applies only to division, not subtraction.

$10^4 \div 10^2 = 10^{4-2} = 10^2 = 100$

$10^4 - 10^2 \neq 10^2$

The easiest way to prove this to yourself: compute the actual values.

$10^4 = 10{,}000$

$10^2 = 100$

$10{,}000 - 100 = 9{,}900$

$9{,}900$ is clearly not $100$. The shortcut produces a wrong answer every time.

The rule: exponent shortcut rules (same base, add or subtract exponents) apply ONLY to multiplication and division. Never to addition or subtraction. When you see addition or subtraction between terms with exponents, compute the actual values.

Step 1: Distribute the Repeating Decimal

Instead of computing $10^4 - 10^2$ first and then multiplying, distribute $0.00\overline{12}$ into the parentheses. This avoids working with $9{,}900$ directly and keeps the decimal shifting clean.

$(10^4 - 10^2) \times 0.00\overline{12} = (10^4 \times 0.00\overline{12}) - (10^2 \times 0.00\overline{12})$

Step 2: Multiply Each Term

Multiplying by $10^4$ (which is $10{,}000$) shifts the decimal four places to the right:

$10^4 \times 0.00\overline{12} = 12.\overline{12}$

The pattern: $0.0012121212...$ becomes $12.12121212...$ when you move the decimal four places right.

Multiplying by $10^2$ (which is $100$) shifts the decimal two places to the right:

$10^2 \times 0.00\overline{12} = 0.\overline{12}$

The pattern: $0.0012121212...$ becomes $0.12121212...$ when you move the decimal two places right.

Step 3: Subtract

Now subtract the two results:

$12.\overline{12} - 0.\overline{12}$

The repeating "$12$" after the decimal point cancels completely. What remains is:

$12.\overline{12} - 0.\overline{12} = 12$

The answer is (E).

Why This Problem Matters

This problem has a 25% miss rate, and the dominant trap answer (B) accounts for nearly half of all errors. That trap comes from one source: misapplying exponent shortcut rules to subtraction.

The fix is a binary rule: if the operation between exponent terms is addition or subtraction, compute the actual values. If the operation is multiplication or division, use the shortcut rules. There's no middle ground.

The secondary lesson is about distribution. You could compute $10^4 - 10^2 = 9{,}900$ and then multiply $9{,}900 \times 0.00\overline{12}$, but that requires multiplying a four-digit number by a repeating decimal. Distributing first and subtracting second keeps the arithmetic cleaner, because each multiplication is just a decimal shift.

The broader habit: when you see parentheses with a multiplier outside, consider whether distributing produces simpler intermediate calculations than computing inside the parentheses first. On this problem, it does.

Want the full strategy behind this problem? Read: GMAT® Quant: Three Systems for Percents, Exponents, and Rounding

From Episode 31 of Real GMAT® Problems (The GMAT® Strategy Podcast).

Ready for the next one? "Cindy Drove Her Car 290 Miles, Rounded to the Nearest 10..." — GMAT® Worked Solution.