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2013 AMC 10A Problem 9

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2013 AMC 10A Problem 9AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20%20\% of her three-point shots and 30%30\% of her two-point shots. Shenille attempted 3030 shots. How many points did she score?

  1. A)

    1212

  2. B)

    1818

  3. C)

    2424

  4. D)

    3030

  5. E)

    3636

Answer

B

Key insight

A three-point attempt yields 0.2 times 3 = 0.6 points and a two-point attempt 0.3 times 2 = 0.6, so every attempt is worth 0.6 regardless of type.

Solution

Let xx be the number of three-point attempts, so there were 30−x30 - x two-point attempts.

Points from three-pointers: 20%20\% of xx attempts succeed, each worth 33, giving 0.2⋅3x=0.6x0.2 \cdot 3x = 0.6x.

Points from two-pointers: 30%30\% of 30−x30 - x attempts succeed, each worth 22, giving 0.3⋅2(30−x)=0.6(30−x)0.3 \cdot 2(30 - x) = 0.6(30 - x).

Total:
0.6x+0.6(30−x)=0.6⋅30=18.0.6x + 0.6(30 - x) = 0.6 \cdot 30 = 18.

The xx cancels, so the split does not matter. Shenille scored (B) 18\boxed{\textbf{(B)}\ 18} points.

Why this works

The problem does not say how many shots of each type were attempted, which is a hint that the answer must not depend on it. Computing points per attempt for each type (0.60.6 and 0.60.6) exposes why: both shot types yield the same average points per attempt, so the total is simply 0.60.6 times the number of attempts.

Alternative approach

Since the answer cannot depend on the split, pick an extreme case: all 3030 shots are three-pointers. Then 20%20\% of 3030 is 66 makes, worth 1818 points.

The trap

Applying 20% and 30% to the number of shots without multiplying by the point values, or assuming the 30 shots split evenly by type.

Common mistakes

  • Applying 20% and 30% to the number of shots without multiplying by the point values, or assuming the 30 shots split evenly by type.
  • Adding the percentages (50%50\% of 3030 shots) and treating that as the number of made shots.

Techniques

Set up the equation/formula and compute; no special trick needed