AMC 10 · 2002 · #1

Grade 8 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
exponentsprime-factorization identify-subproblems ↑ Prerequisites: exponents
📏 Short solution 💡 1 insight
Problem
The numerator 2²⁰⁰¹ · 3²⁰⁰³ is a power of 2 times a power of 3, and the denominator is 6²⁰⁰². Simplify the fraction (2²⁰⁰¹ · 3²⁰⁰³)/6²⁰⁰² to a single value.

Pick an answer.

(A)
1/6
(B)
1/3
(C)
1/2
(D)
2/3
(E)
3/2

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The one obstacle is that the denominator's base 6 does not match the numerator's bases 2 and 3. Tool #7 (Identify Subproblems) fixes this by splitting 6²⁰⁰² into a base-2 piece and a base-3 piece, turning one scary fraction into two clean same-base quotients. Tool #9 (Solve an Easier Related Problem) then makes the giant exponents harmless: because the powers differ by only 1, we subtract exponents instead of ever evaluating 2²⁰⁰¹ or 3²⁰⁰³. Tool #3 (Eliminate Possibilities) guards the two built-in traps — subtracting the exponents the wrong way gives (D) 2/3, and dropping the leftover 3 gives (C) 1/2.

1STEP 1

Break the base 6 apart

Base 6 shares nothing with the numerator, so split it: 6²⁰⁰²=(2 · 3)²⁰⁰²=2²⁰⁰² · 3²⁰⁰². Now every base is 2 or 3.

6²⁰⁰²=(2 · 3)²⁰⁰²=2²⁰⁰² · 3²⁰⁰²
2STEP 2

Group the 2s and the 3s

Rewrite with like bases together: (2²⁰⁰¹ · 3²⁰⁰³)/(2²⁰⁰² · 3²⁰⁰²)=2²⁰⁰¹/2²⁰⁰² · 3²⁰⁰³/3²⁰⁰² — two tidy same-base fractions.

(2²⁰⁰¹ · 3²⁰⁰³)/(2²⁰⁰² · 3²⁰⁰²)=2²⁰⁰¹/2²⁰⁰²·3²⁰⁰³/3²⁰⁰²
3STEP 3

Subtract exponents on each base

Same base divides by subtracting exponents: 2²⁰⁰¹/2²⁰⁰²=2⁻¹=1/2 and 3²⁰⁰³/3²⁰⁰²=3¹=3. Only the gap of 1 matters.

2²⁰⁰¹/2²⁰⁰²=2⁻¹=1/2, 3²⁰⁰³/3²⁰⁰²=3¹=3
4STEP 4

Multiply the two pieces

Multiply the pieces: 1/2 · 3=3/2, choice (E). Reversing the subtraction gives (D); dropping the leftover 3 gives (C).

1/2 · 3=3/2 → (E)
Answer
3/2
A quick sanity test on tiny exponents confirms the pattern: (2¹ · 3³)/6²=(2 · 27)/36=54/36=3/2, the same value. That makes sense because the base-2 exponent sits one below the shared power (giving 2⁻¹) while the base-3 exponent sits one above it (giving 3⁺¹), so the answer is 3/2 regardless of how large the shared exponent is. A value just above 1 is reasonable, and it rules out the small fractions (A)–(D).
💡Key takeaway

Break every base down to primes so they match, then dividing same-base powers is just subtracting the exponents — the giant numbers never need to be worked out.

  • Break the base 6 apart
  • Group the 2s and the 3s
  • Subtract exponents on each base
  • Multiply the two pieces

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