AMC 10 · 2004 · #11
Grade 7 algebraPick an answer.
The five 100s drag the average up, and the only thing that can drag it back down to 76 is a supply of low scores. Tool #4 (Introduce a Variable) names the class size n and turns the mean into a fixed total. Tool #14 (Extreme Principle) asks how far down the other students can possibly pull the total: they are floored at 60, so there is a hard limit, and that limit becomes an inequality in n. Tool #13 (Convert to Algebra) solves it. The inequality only rules small classes out, so Tool #6 (Guess and Check) finishes by exhibiting one honest score list for the surviving size.
Turn the mean into a total
The average becomes a total, which is far easier to control.
An average is a total in disguise: fix the class size and the average pins the total exactly.
Fixing the class size turns the average into an exact total the scores must add up to.
▸ Why?
An average is the total shared out equally, so multiplying it back by the count returns the total.
▸ Why?
That total is the individual scores added together, so pushing some scores to their floor bounds all the rest.
Push everyone else to the floor
Pushing everyone else to the floor gives the smallest possible total.
To see whether a low average is reachable, imagine everyone who is allowed to score low scoring as low as the rules permit.
7.EE.A.1Extreme PrincipleCompare the floor with the required total
The floor must not exceed what is required, forcing at least 13.
The low scorers can only pull the average down so far, and that limit tells you how many of them you need.
7.EE.B.4Convert To AlgebraBuild a class of 13 that works
A class of 13 really exists with eight scores of 61, so the answer is 13, choice (D).
A bound tells you what is impossible; only an example tells you what is possible, so always build the example.
7.NS.A.3Guess And CheckFive perfect scores push the average up, and only students sitting at the 60-point floor can pull it back down, so count how many floor-sitters you need and then actually build the class to prove that size works.
- Turn the mean into a total
- Push everyone else to the floor
- Compare the floor with the required total
- Build a class of 13 that works