AMC 10 · 2008 · #4
Grade 5 algebraPick an answer.
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.
Multiply the first few factors
Multiplying the first few factors shows a pattern.
Trying a tiny version of the product shows what happens without doing all the work.
5.NF.B.4Solve An Easier Related ProblemSee why a number always cancels
The tops and bottoms are the same list, shifted by one.
Because each numerator equals the next denominator, matching pairs sit on top and bottom ready to cancel.
4.OA.C.5Look For A PatternCancel everything in the middle
So everything cancels but the two ends.
A number on top and the same number on the bottom multiply to 1, so only the two unmatched ends are left.
Each number on top meets the same number on the bottom, so only the two unmatched ends survive.
▸ Why?
A number divided by itself undoes the multiplication, leaving the product untouched.
▸ Why?
Multiplying by one changes nothing, so every cancelled pair simply drops out of the chain.
Do the final division
Dividing gives 502, choice (B).
Once the chain telescopes, a single division finishes the problem.
4.NBT.B.6Identify SubproblemsWhen 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