AMC 10 · 2022 · #2
Grade 6 algebraPick an answer.
All three numbers can be described from the third: first = 6z, second = z + 40. That makes the third number a single dial we can turn. Tool #6 (Guess and Check) lets us test a small whole number for z and see if the total is 96 — much friendlier than naming three variables. Tool #11 (Work Backwards) and #7 (Subproblems) take over once z is found: plug it back to recover the first and second numbers, then compute their absolute difference.
Write all three with one letter
Write all three with one letter.
Naming the smallest piece z and writing the others in terms of it is the Grade 6 "use a variable to stand for the unknown" idea — one number controls all three.
Naming the smallest piece and writing the others from it turns three unknowns into one.
▸ Why?
Each of the others is a fixed multiple or shift of that one, so they all ride on it.
▸ Why?
Their differences are fixed no matter what that one number turns out to be.
Use the sum
The sum gives one equation.
Two directed guesses sandwich z between 5 and 10; one more lands on 7 — Grade 6 "solve px+q=r" without needing pencil-and-paper algebra.
6.EE.B.7Guess And CheckFind the two numbers
Recover both numbers from the letter.
Working back from z = 7 through the two simple expressions gives the actual numbers — Grade 5 "evaluate a numerical expression".
5.OA.A.1Work BackwardsTake the difference
The absolute difference is 5.
Absolute value is just the distance between two numbers — Grade 6 "ordering and absolute value".
6.NS.C.7Work BackwardsThis AMC 12 problem only needs Grade 6 "name the unknown and try small numbers" — once the third number turns out to be 7, the gap between 42 and 47 is just 5.