AMC 10 · 2008 · #5

Grade 5 algebra
Archetype Telescoping Collapse └ Sub-archetype Product Ratio Cancellation
fraction-multiplicationpattern-recognitiontelescoping-sum pattern-recognitioneasier-related-problem ↑ Prerequisites: fraction-multiplication
📏 Medium solution 💡 2 insights
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Problem
A long chain of fractions is multiplied: 8/4·12/8·16/12…(4n+4)/4n…2008/2004. Every numerator is exactly 4 more than its own denominator, and the chain ends at 2008/2004. Find which answer choice the whole product equals.

Pick an answer.

(A)
251
(B)
502
(C)
1004
(D)
2008
(E)
4016

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Multiplying 501 fractions head-on is hopeless, so Tool #9 (Solve an Easier Related Problem) says try just the first two or three factors first. That reveals a repeating cancellation, and Tool #5 (Look for a Pattern) explains it: the top of each fraction is the bottom of the next one. Tool #7 (Identify Subproblems) then lets me write the whole thing as one big fraction, cancel every matching pair, and finish with a single easy division.

1STEP 1

Multiply the first few factors

Try just two factors: 8/4·12/8=12/4=3. Bring in the next one and the 12 cancels too, leaving 16/4=4.

8/4·12/8=12/4=3, 12/4·16/12=16/4=4
2STEP 2

See why a number always cancels

Tops run 8, 12, 16, …, 2008 and bottoms run 4, 8, 12, …, 2004, so each numerator is the next fraction's denominator.

tops: 8,12,16,…,2008; bottoms: 4,8,12,…,2004
3STEP 3

Cancel everything in the middle

Write it as one fraction: (8·12·16…2008)/(4·8·12…2004). Everything from 8 to 2004 appears twice and cancels, leaving 2008/4.

(8·12·16…2008)/(4·8·12…2004)=2008/4
4STEP 4

Do the final division

Only 2008/4 is left, and 2008÷4=502 — that is answer choice (B).

2008/4=502 → (B)
Answer
502
Check the division: 502·4=2008, so 2008/4=502 is right. The size is sensible too — each factor is only a little bigger than 1 (about 2, 1.5, 1.33, …), so multiplying about 500 of them lands well below the largest choices. The traps line up with common slips: 2008 (D) forgets to divide by the first denominator, 4016 (E) doubles instead, and 251 (A) is 2008/8 from cancelling one number too many.
💡Key takeaway

When the top of each fraction is the bottom of the next, they cancel down the line, so only the very last top and the very first bottom are left.

  • Multiply the first few factors
  • See why a number always cancels
  • Cancel everything in the middle
  • Do the final division

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