AMC 10 · 2024 · #18
Grade 5 arithmeticPick an answer.
The problem hands us a finite, well-defined sum of 10 ratios. Tool #2 (Make a Systematic List) is the workhorse: list F₁, …, F₂₀ in order, then list the 10 ratios. Tool #9 (Solve an Easier Related Problem) gets us moving even before that — compute the first three or four ratios by hand and notice they are integers (1, 3, 4, 7, …). Tool #5 (Look for a Pattern) then turns those into a Fibonacci-style recurrence (a_n = a_n-1 + a_n-2 starting 1, 3), so the rest of the list extends by addition rather than by dividing big Fibonacci numbers. With 10 integers in hand, the answer is one careful sum.
List the terms you need
Write out the first twenty terms.
Grade 4 "generate a number pattern given a rule" — start with 1, 1 and add.
4.OA.C.5Make A Systematic ListCompute the first few quotients
They come out 1, 3, 4, 7, 11.
Grade 5 fraction-as-division: each F₂n/F_n is a clean small integer because F_n divides F₂n exactly.
Each of these quotients comes out a clean whole number, because the smaller term divides the bigger one exactly.
▸ Why?
A quotient is a whole number exactly when the division leaves no remainder.
▸ Why?
The smaller term's whole prime recipe sits inside the bigger one, so nothing is left uncancelled.
Spot the rule
Each quotient is the sum of the previous two.
Grade 4 number-pattern rule: compute four or five terms, conjecture, verify with one more — "4 = 1 + 3, 7 = 3 + 4" is the standard tell.
4.OA.C.5Look For A PatternContinue the list
Next come 18, 29, 47, 76, 123.
Grade 4 pattern-extension by addition replaces five tedious Fibonacci divisions.
4.OA.C.5Make A Systematic ListAdd the ten terms
Their total is 319.
Grade 4 multi-digit addition done in small chunks — partial sums grow steadily, no big number tricks needed.
4.NBT.B.4Make A Systematic ListHard-looking Fibonacci ratios become a simple pattern game once you compute the first few: 1, 3, 4, 7, 11 extends by addition to 123, and ten numbers sum to 319 — just Grade 5 division and Grade 4 number-pattern rules.