AMC 10 · 2002 · #1
Grade 8 rate-ratioPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The fraction looks frightening only because of the giant exponents. Tool #7 (Identify Subproblems) splits the job into three clean pieces: factor the numerator, collapse the denominator, then cancel. Tool #9 (Solve an Easier Related Problem) is what makes the exponents harmless — pulling the common factor 10²⁰⁰⁰ out of the top and bottom removes every huge power at once, leaving a tiny fraction of small whole numbers. Tool #3 (Eliminate Possibilities) then handles 'closest to': once the value is 5.05, the choices 0.1, 0.2, 1, and 10 are all far away, so only 5 survives.
Factor the numerator
Both top terms are powers of 10, so pull out the smaller one: 10²⁰⁰⁰(1+10²)=101 · 10²⁰⁰⁰.
Adding powers of the same base lets you factor out the smallest power, shrinking the exponents into an ordinary number.
Adding powers of the same base lets you factor out the smallest power, shrinking the exponents to an ordinary number.
▸ Why?
A factor shared by every term can be lifted out of the sum, leaving something small inside.
▸ Why?
An exponent counts how many times the base is used, so the smaller power sits inside the larger one as a factor.
Collapse the denominator
The bottom is one thing added to itself, and 10²⁰⁰¹=10 · 10²⁰⁰⁰, so it is 2 · 10²⁰⁰¹=20 · 10²⁰⁰⁰.
Anything plus itself is just twice that thing, and one extra factor of 10 turns 10²⁰⁰¹ into 10 · 10²⁰⁰⁰.
6.EE.A.3Identify SubproblemsCancel the common power of 10
In (101 · 10²⁰⁰⁰)/(20 · 10²⁰⁰⁰) the shared factor 10²⁰⁰⁰ cancels completely, leaving 101/20.
A factor shared by numerator and denominator divides out to 1, no matter how enormous it is.
8.EE.A.1Solve An Easier Related ProblemDivide and pick the closest choice
101/20=5.05, and 0.1, 0.2, 1, and 10 are all far off, so the closest choice is 5 — that is (D).
Once the monster fraction reduces to 101/20, ordinary division settles which choice is nearest.
5.NBT.B.7Eliminate PossibilitiesWhen every term is a power of the same base, factor out the smallest power and cancel it — the giant exponents disappear and a tiny fraction is left.
- Factor the numerator
- Collapse the denominator
- Cancel the common power of 10
- Divide and pick the closest choice