AMC 10 · 2022 · #1
Grade 5 arithmeticPick an answer.
The expression has three nested layers, so we cannot evaluate it left-to-right. Tool #7 (Identify Subproblems) splits the giant fraction into three identical small jobs: "compute 3 + 1/(something already known)". Tool #11 (Work Backwards) tells us where to start — at the deepest layer, then climb outward, replacing each □ with the value we just found. Algebra (#13) is overkill since there are no unknowns; the layered structure tells us the path.
Evaluate the innermost level
Start from the innermost level.
Adding a whole number and a unit fraction is the standard Grade 5 "unlike denominators" job — rename 3 as 9/3 and add.
5.NF.A.1Work BackwardsTake the reciprocal
A reciprocal is just a flip.
"1 divided by a fraction" is the same as flipping that fraction — Grade 5 interpretation of a fraction as a division.
One divided by a fraction is the same as flipping that fraction over.
▸ Why?
A number multiplied by its reciprocal gives one, so the flip is exactly what undoes it.
▸ Why?
Turning a fraction over swaps its top and bottom without changing what it counts.
Move out one level
Add three to move out a level.
Same Grade 5 "common denominator" trick — the middle layer collapses to a tidy fraction.
5.NF.A.1Work BackwardsFinish the last level
Finishing gives one hundred nine over thirty-three.
One last "flip and add over a common denominator" finishes the climb — pure Grade 5 fraction work.
5.NF.A.1Identify SubproblemsThis AMC 12 problem only needs Grade 5 "add fractions with unlike denominators and flip a fraction" — work from the deepest 1/3 outward, one layer at a time.