Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #19
Grade 6 rate-ratioPick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
One sentence asks for sticks of butter, but four short jobs are hiding inside it — the classic signal for Tool #7 (Break into Subproblems). The chain is: (i) total cookies needed, (ii) number of pans (round up to a whole pan), (iii) total tablespoons of butter, (iv) number of sticks (round up to a whole stick). Each subproblem is a single arithmetic move, and the two rounding-up steps come from the real-world constraints "full recipes only" and "whole sticks only." No algebra is required; tracking the units carries the whole solution.
Find the cookies needed
Subproblem 1: multiply the 108 students by 2 cookies each to get 216 cookies needed.
"Average of 2 apiece" times the number of students gives the total — a Grade 4 multistep word-problem setup.
4.OA.A.3Identify SubproblemsFind the pans needed
Subproblem 2: 216 ÷ 15 = 14.4, but full recipes only — round up to 15 pans (14 pans give just 210).
Real-world "must be at least this many, no halves allowed" forces a round-up, not the usual nearest-whole-number rounding.
6.NS.B.2Identify SubproblemsFind the tablespoons of butter
Subproblem 3: each of the 15 pans uses 3 tablespoons, so 15 × 3 = 45 tablespoons of butter.
Same multiplicative step as subproblem 1: amount per unit times number of units.
5.NBT.B.5Identify SubproblemsConvert tablespoons to sticks
Subproblem 4: 45 ÷ 8 = 5.625, and partial sticks can't be bought — round up to 6 sticks (5 sticks give only 40).
Same "at least this many, whole units only" constraint as the pans step — round up, even when the fractional part is small.
With 45 tablespoons of butter required and butter sold only in whole 8-tablespoon sticks, the count to buy is the smallest whole number of sticks whose total still reaches 45 tablespoons.
▸ Why?
Buying a whole number n of sticks supplies exactly 8 × n tablespoons of butter.
▸ Why?
A stick is a larger unit that always equals the same fixed 8 smaller tablespoon units, so one stick converts to 8 tablespoons.
▸ Why?
n sticks are n equal groups of 8 tablespoons, and repeatedly adding the same 8 is exactly what multiplying 8 by n means.
▸ Why?
Because each added stick raises the total by the same fixed 8 tablespoons, the supply climbs in equal jumps, so exactly one whole-stick count is the first to reach 45 while every smaller whole count falls short.
▸ Why?
Each extra stick adds one more equal group of 8 tablespoons, so the totals rise in steady, countable steps toward the 45 needed.
Word problems with real-life "whole units only" constraints become a chain of small jobs. Multiply, divide, round UP whenever a partial unit is not allowed — that turns a wordy AMC 8 problem into four short Grade 4-6 arithmetic steps.
- Find the cookies needed
- Find the pans needed
- Find the tablespoons of butter
- Convert tablespoons to sticks
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