AMC 10 · 2018 · #14
Grade 8 arithmeticPick an answer.
AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #4 (Introduce a Variable): the exponents only differ by 4, so pulling out the common powers and naming a=3⁹⁶ and b=2⁹⁶ collapses the monster fraction into the tidy expression (81a+16b)/(a+b) that I can actually manipulate. Tool #9 (Solve an Easier Related Problem): I never need the true gigantic value — I only need to trap it between two integers, so I replace exact arithmetic with a simple bound on one small leftover term. Tool #3 (Eliminate Possibilities): once I show the value sits strictly between 80 and 81, four of the five choices are impossible and only one survives.
Pull out the common powers
Peel off 3⁹⁶ and 2⁹⁶ and name a=3⁹⁶, b=2⁹⁶; since 3¹⁰⁰=81· 3⁹⁶ and 2¹⁰⁰=16· 2⁹⁶, the fraction becomes (81a+16b)/(a+b).
Same base, four steps apart, means each top term is just the bottom term times 3⁴ or 2⁴.
8.EE.A.1Use Matrix LogicRewrite to expose 81
Compare the top to 81(a+b): it falls short by 65b, so 81a+16b=81(a+b)-65b and the fraction splits into 81-65b/(a+b).
Writing it as 81 minus a leftover turns a hard fraction into 'a little below 81.'
7.EE.A.2Use Matrix LogicThe value is below 81
Both a and b are positive, so 65b/(a+b) is a positive amount subtracted from 81, forcing the value strictly below 81.
Take a positive bite out of 81 and you land somewhere under 81.
6.NS.C.7Use Matrix LogicThe leftover is tiny
Since a+b ＞ a, 65b/(a+b) ＜ 65(2/3)⁹⁶, and 3⁹⁶ dwarfs 65· 2⁹⁶ so this is less than 1, keeping the value above 80.
3⁹⁶ towers over 65· 2⁹⁶, so the bite taken out of 81 is way smaller than 1.
7.NS.A.3Solve An Easier Related ProblemTrap between 80 and 81
The value is under 81 but over 80, so it sits between consecutive integers, making its floor 80 — the answer is (A).
A number stuck between 80 and 81 has floor exactly 80.
6.NS.C.7Eliminate PossibilitiesPull out the shared powers so the fraction becomes 81 minus a tiny leftover; the leftover is less than 1, so the value is squeezed between 80 and 81 and its floor is 80.
- Pull out the common powers
- Rewrite to expose 81
- The value is below 81
- The leftover is tiny
- Trap between 80 and 81