Competition · AMC preparation · step 4 of 4
AMC 8 · 2006 · #9
Grade 5 arithmeticPick an answer.
AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Try a shorter product
Try a shorter version first: multiply just the first three factors into one fraction.
Trying a shorter version is the Grade 5 "multiply fractions" move — small enough to do by hand, big enough to see what happens.
5.NF.B.4Solve An Easier Related ProblemSee what cancels
The 3 and 4 cancel top and bottom, so only the first denominator 2 and last numerator 5 survive: .
Spotting that interior numbers appear once on top and once on bottom is the Grade 4 "generate and identify patterns" move.
4.OA.C.5Look For A PatternApply the pattern to all
Apply the same pattern to every factor: all interior numbers from 3 to 2005 cancel, leaving .
The cancellation rule extends from the short product to the long one because nothing structural changes — only the length.
Multiplying all the factors together makes every number from 3 to 2005 disappear, leaving the product equal to 2006/2.
▸ Why?
Multiplying the fractions puts every top number into one big numerator and every bottom number into one big denominator, and these factors can be lined up so each top number sits beside the equal number on the bottom.
▸ Why?
Changing the order in which you multiply the factors does not change the product, so each numerator can be moved next to the matching denominator.
▸ Why?
Changing which factors you group and combine first does not change the product, so each matching top-and-bottom pair can be combined on its own.
▸ Why?
Each matched pair is one interior number over that same number, and a number divided by itself is one, because one times the number gives the number back.
▸ Why?
Every matched pair has turned into one, and multiplying by one leaves a value unchanged, so all the pairs drop out and only the unmatched first bottom number 2 and last top number 2006 remain.
Divide to finish
Divide to finish.
A single Grade 5 division turns the telescoped fraction into the answer.
5.NBT.B.6Look For A PatternWhen 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.
- Try a shorter product
- See what cancels
- Apply the pattern to all
- Divide to finish
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