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2013 AMC 10B Problem 10

2013 AMC 10B Problem 10AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?

A)
$10$
B)
$15$
C)
$20$
D)
$25$
E)
$30$

Answer

Key insight

Let x be three-point attempts; two-point attempts are 1.5x, and points are 2(0.5)(1.5x) + 3(0.4)x = 2.7x = 54.

Solution

Let $x$ be the number of three-point attempts. "50% more" two-point attempts means $1.5x$ of them.

Points from two-pointers: half of $1.5x$ attempts succeed, each worth $2$ , giving $2 \cdot 0.5 \cdot 1.5x = 1.5x$ points.

Points from three-pointers: $40\%$ of $x$ succeed, each worth $3$ , giving $3 \cdot 0.4 \cdot x = 1.2x$ points.

Total: $1.5x + 1.2x = 2.7x = 54$ , so $x = 20$ .

Check: $30$ two-point attempts, $15$ made, $30$ points; $20$ three-point attempts, $8$ made, $24$ points; $30 + 24 = 54$ .

The answer is $\boxed{\textbf{(C)}\ 20}$ .

Why this works

Every quantity in the problem is a fixed multiple of the number of three-point attempts, so a single variable suffices. Translate each percent into a multiplier before writing the equation: attempts $\to$ made shots $\to$ points is a chain of multiplications.

Alternative approach

Test the choices: each three-point attempt is worth $1.2$ points on average and brings $1.5$ two-point attempts worth $1.5$ points, so each unit of $x$ yields $2.7$ points; $54 / 2.7 = 20$ .

The trap

Reading '50% more two-point shots' as 50% of x or as 2x, or forgetting to multiply made shots by their point values.

Common mistakes

Reading '50% more two-point shots' as 50% of x or as 2x, or forgetting to multiply made shots by their point values.
Reporting the number of two-point attempts ( $30$ , choice (E)) instead of three-point attempts.

Techniques

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