AMC 10 · 2013 · #9
Grade 6 arithmeticIn a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score?
Pick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Shenille took 30 shots, some worth 3 points and some worth 2 points. She made 20% of her three-point shots and 30% of her two-point shots. Find her total score.
Givens: Every shot was either a three-point attempt or a two-point attempt.; She made 20% of the three-point attempts, each made one worth 3 points.; She made 30% of the two-point attempts, each made one worth 2 points.; The two kinds of attempts together add up to 30 shots.
Unknowns: The total number of points Shenille scored.
Understand
Restated: Shenille took 30 shots, some worth 3 points and some worth 2 points. She made 20% of her three-point shots and 30% of her two-point shots. Find her total score.
Givens: Every shot was either a three-point attempt or a two-point attempt.; She made 20% of the three-point attempts, each made one worth 3 points.; She made 30% of the two-point attempts, each made one worth 2 points.; The two kinds of attempts together add up to 30 shots.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #7 Identify Subproblems
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.
Execute — Answer: B
6.EE.A.2 Step 1 Name the two unknown counts
- Let x be the number of three-point attempts and y the number of two-point attempts.
- Since every one of the 30 shots is one type or the other, the two counts add to 30.
💡 Giving the two unknown counts names lets you work with them even though neither is known yet.
6.RP.A.3 Step 2 Points from each shot type
- She makes 20% of the x three-point attempts, and each made shot is worth 3 points, giving 0.20 times x times 3.
- She makes 30% of the y two-point attempts, each worth 2 points, giving 0.30 times y times 2.
- Both simplify to 0.6 times the count.
💡 A lower success rate on the higher-value shot balances out to the same 0.6 points per attempt for both types.
6.EE.A.3 Step 3 Add and factor out the common 0.6
- The total score is the sum of the two parts, 0.6x plus 0.6y.
- Both terms share the factor 0.6, so pull it out to get 0.6 times (x plus y).
💡 Because both shot types earn 0.6 per attempt, only the total number of attempts matters, not how they split.
6.EE.A.2 Step 4 Substitute the known total
- Replace x plus y with 30, since that is the total number of attempts.
- Multiplying gives 0.6 times 30, which equals 18 points.
- That matches choice (B).
💡 Once the split cancels, the answer depends only on the 30 total attempts you already know.
6.EE.A.2 Let x be the number of three-point attempts and y the number of two-point attemp 6.RP.A.3 She makes 20% of the x three-point attempts, and each made shot is worth 3 point 6.EE.A.3 The total score is the sum of the two parts, 0.6x plus 0.6y. Both terms share th 6.EE.A.2 Replace x plus y with 30, since that is the total number of attempts. Multiplyin Review
Reasonableness: If Shenille had made every shot, 30 attempts could yield at most 90 points, but she made only 20% to 30% of them, so a score far below that is expected. 18 points is a small fraction of 90 and sits near the low end of the choices, which fits her low success rates.
Alternative: Use the extreme principle: since the split does not matter, assume all 30 were three-pointers. She makes 20%, or 6 shots, worth 3 points each: 6 times 3 = 18. Assuming all 30 were two-pointers gives 30% of 30 = 9 made, worth 2 each: 9 times 2 = 18. Either extreme gives 18 (B).
CCSS standards used (min grade 6)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Naming the attempt counts x and y and evaluating the score expression at x + y = 30.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Taking 20% and 30% of the attempt counts to find made shots and their points.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Factoring 0.6x + 0.6y into 0.6(x + y) so the unknown split cancels.)
⭐ Both kinds of shot earn the same 0.6 points per attempt, so just multiply 0.6 by all 30 attempts.
⭐ Both kinds of shot earn the same 0.6 points per attempt, so just multiply 0.6 by all 30 attempts.
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