"3.003 ÷ 2.002 =..." — GMAT® Worked Solution

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

$\dfrac{3.003}{2.002} =$

(A) $1.05$

(B) $1.50015$

(C) $1.501$

(D) $1.5015$

(E) $1.5$

Try it before reading on.

Approach 1: Factoring (the faster path)

Before reaching for long division, scan the numbers for structural similarity. Look at $3.003$ and $2.002$:

• Both have a single-digit whole number followed by a three-digit decimal
• Both have the same decimal pattern: $.00X$ where $X$ matches the whole number
• This isn't a coincidence. It signals that both numbers can be factored into a single-digit multiplier and a common factor.

Rewrite each number as a product:

$3.003 = 3 \times 1.001$

$2.002 = 2 \times 1.001$

Now substitute these factored forms back into the original fraction:

$\dfrac{3.003}{2.002} = \dfrac{3 \times 1.001}{2 \times 1.001}$

The $1.001$ appears in both numerator and denominator, so it cancels:

$\dfrac{3 \times 1.001}{2 \times 1.001} = \dfrac{3}{2} = 1.5$

The answer is (E).

Approach 2: Long Division (the reliable fallback)

If the factoring pattern doesn't jump out, long division works every time. Set up $3.003 \div 2.002$:

      1.5
     -----
2.002 ) 3.003

First, shift the decimal in both numbers to make the divisor a whole number. Move three places right: $3003 \div 2002$.

        1.5
       -----
  2002 ) 3003.0
        2002
        ----
         1001 0
         1001 0
         -----
             0

$2002$ goes into $3003$ once ($2002$), remainder $1001$. Bring down the $0$, making $10010$. $2002$ goes into $10010$ five times ($10010$), remainder $0$.

Result: $1.5$.

The answer is (E).

How to Recognize the Factoring Pattern

The factoring approach is faster, but it requires recognizing the pattern. Here's what to look for:

Matching digit structure. If two numbers in a fraction have the same number of digits, the same decimal placement, and similar digit patterns, they probably share a common factor. $3.003$ and $2.002$ both have a one-digit whole number and a three-digit decimal where the decimal digits echo the whole number.

Scaled versions of a common base. $3.003 = 3 \times 1.001$ and $2.002 = 2 \times 1.001$. The $1.001$ is the shared structure. When you see numbers that look like one digit multiplied by a repeating or mirrored number, try factoring out the common piece.

When to check. Spend about two seconds scanning the numbers before starting any computation. If the digit patterns match, explore factoring. If they don't, go straight to long division. The two-second scan costs almost nothing, and when it works, it saves a minute or more.

Why This Problem Matters

About 27% of test takers miss this problem, making it the hardest of the three in this episode by the numbers.

The wrong answers split between (B) $1.50015$ at 10% and (D) $1.5015$ at 9%. Both look like partial results from long division where a calculation error crept in partway through. They also look like what you might get if you tried to divide the numbers visually, splitting $3 \div 2 = 1.5$ and $.003 \div .002 = 1.5$ and combining them somehow.

The visual-division approach reveals a misunderstanding of how decimal arithmetic works. You can't split a fraction into whole-number and decimal parts and divide them separately. Decimal division doesn't work that way. If you were tempted by (B) or (D), practicing decimal long division is the highest-value fix.

The broader lesson: long division is a sure thing if you execute it correctly, but it's slow and error-prone if you haven't practiced. Factoring is faster and cleaner when you can spot the pattern, but it requires pattern recognition that comes from exposure. Practice both approaches on every fraction problem you see. Over time, the pattern recognition becomes automatic, and you'll know within two seconds which approach to use.

If you're filing problems for future re-solve, this one is worth coming back to in a few weeks. Try the factoring approach the second time around and see if the pattern jumps out faster. If it does, the habit is building.

Want the full strategy behind this problem? Read: GMAT® Quant Fundamentals: Process Over Computation

From Episode 33 of Real GMAT® Problems (The GMAT® Strategy Podcast).