AMC 8 · 2006 · #9

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
fraction-multiplicationpattern-recognitionfraction-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: fraction-arithmeticfraction-multiplication
📏 Short solution 💡 2 insights
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Problem
Find the value of the product 32\frac{3}{2}×43\frac{4}{3}×54\frac{5}{4}×…×20062005\frac{2006}{2005}, where each factor is a fraction whose numerator is one more than its denominator and whose denominator matches the previous factor's numerator.

Pick an answer.

(A)
1
(B)
1002
(C)
1003
(D)
2005
(E)
2006

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Multiplying out 2004 fractions directly is impossible, so the problem is asking us to spot structure. Tool #5 (Look for a Pattern) is the natural primary: each factor's numerator matches the next factor's denominator, which is the signature of a telescoping product where most terms cancel. To make the cancellation visible without writing all 2004 factors, Tool #9 (Try an Easier Problem) is the perfect helper — first compute the same product for a few terms, see what survives, and trust the pattern to extend.

1STEP 1

Try a shorter version first: multiply just the first three factors into one fraction.

32\frac{3}{2}×43\frac{4}{3}×54\frac{5}{4} = 3×4×52×3×4\frac{3× 4× 5}{2× 3× 4} = 52\frac{5}{2}
2STEP 2

The 3 and 4 cancel top and bottom, so only the first denominator 2 and last numerator 5 survive: 52\frac{5}{2}.

32\frac{3}{2}×43\frac{4}{3}×54\frac{5}{4} = 3×4×52×3×4\frac{3×4× 5}{2×3×4} = 52\frac{5}{2}
3STEP 3

Apply the same pattern to every factor: all interior numbers from 3 to 2005 cancel, leaving 20062\frac{2006}{2}.

32\frac{3}{2}×43\frac{4}{3}×54\frac{5}{4}×…×20062005\frac{2006}{2005} = 20062\frac{2006}{2}
4STEP 4

Divide to finish.

20062\frac{2006}{2} = 1003 → (C)
Answer
1003
Check the pattern against one more easy case. The product 32\frac{3}{2}×43\frac{4}{3}×54\frac{5}{4}×65\frac{6}{5} should leave 62\frac{6}{2} = 3. Multiplying directly: 3⋅4⋅5⋅62⋅3⋅4⋅5\frac{3· 4· 5· 6}{2· 3· 4· 5} = 360120\frac{360}{120} = 3. Match. The rule "answer = last numerator ÷ first denominator" is confirmed, and applying it to the contest product gives 20062\frac{2006}{2} = 1003. Also, the answer must be slightly larger than 1000 since each factor is just above 1 but there are 2004 of them; 1003 fits while 1, 2005, and 2006 do not.
💡Key takeaway

When every fraction's top matches the next fraction's bottom, almost everything cancels. Only the first bottom and the last top survive — here that gives 2006 ÷ 2 = 1003.

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