AMC 10 · 2013 · #6
Grade 6 rate-ratioPick an answer.
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
The two counts add to the known total.
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
Each kind's points share the same factor.
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
Factoring makes the split vanish.
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 points per attempt, only the total number of attempts matters.
▸ Why?
A shared factor can be lifted out of the sum, leaving just the total count inside.
▸ Why?
With the points per attempt fixed, the total rises in step with the attempts alone.
Substitute the known total
The total is 18, 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