AMC 10 · 2013 · #9
Grade 6 arithmeticPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The split between three-point and two-point attempts is unknown, so name each count with a letter and write the score as an expression. Handling the points from each shot type as its own subproblem shows that both types give the same points per attempt, which makes the unknown split cancel out.
Name the two unknown counts
Let x be the number of three-point attempts and y the number of two-point attempts; every shot is one or the other, so x + y = 30.
Giving the two unknown counts names lets you work with them even though neither is known yet.
6.EE.A.2Introduce A VariablePoints from each shot type
A three-point attempt averages 0.20 times 3 points and a two-point attempt 0.30 times 2 — both come to 0.6 points per attempt.
A lower success rate on the higher-value shot balances out to the same 0.6 points per attempt for both types.
6.RP.A.3Identify SubproblemsAdd and factor out the common 0.6
The score is 0.6x + 0.6y, and both terms share the factor 0.6, so pull it out to get 0.6(x + y).
Because both shot types earn 0.6 per attempt, only the total number of attempts matters, not how they split.
Because both shot types earn the same amount per attempt, only the total number of attempts matters.
▸ Why?
Each attempt is worth one fixed amount on average, so the value attaches to the attempt itself.
▸ Why?
A factor shared by both terms can be lifted out, leaving just the sum of the attempts.
Substitute the known total
Since x + y is the total number of attempts, 30, the score is 0.6 times 30 = 18 points, choice (B).
Once the split cancels, the answer depends only on the 30 total attempts you already know.
6.EE.A.2Introduce A VariableBoth kinds of shot earn the same 0.6 points per attempt, so just multiply 0.6 by all 30 attempts.
- Name the two unknown counts
- Points from each shot type
- Add and factor out the common 0.6
- Substitute the known total