AMC 10 · 2013 · #10

Grade 7 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

systems-of-equationspercentagelinear-equations-two-var convert-to-algebra ↑ Prerequisites: linear-equations-two-varpercentage

📏 Medium solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: A team scores 54 points from two-point and three-point shots. They make 50% of their two-point tries and 40% of their three-point tries, and they take 50% more two-point tries than three-point tries. Find how many three-point shots they attempted.

Givens: 50% of two-point shots attempted are made; 40% of three-point shots attempted are made; Total points scored is 54; Two-point attempts are 50% more than three-point attempts

Unknowns: The number of three-point shots attempted

Understand

Givens: 50% of two-point shots attempted are made; 40% of three-point shots attempted are made; Total points scored is 54; Two-point attempts are 50% more than three-point attempts

Plan

Primary tool: #4 Introduce a Variable

Secondary: #13 Convert to Algebra, #8 Analyze the Units

Two unknown counts are linked by a scoring total and a ratio. Name each count with a letter, translate every sentence into an equation, and track the point units so the totals line up. Then substitution collapses it to one equation in one variable.

Execute — Answer: C

#4 Introduce a Variable 6.EE.B.6 Step 1

Name the two counts

Let x be the number of two-point shots attempted and y be the number of three-point shots attempted.
These are the two quantities everything else is described in terms of.

$$x = \text{two-point attempts}, \quad y = \text{three-point attempts}$$

💡 Give each unknown a letter so the word sentences can become equations.

#13 Convert to Algebra 7.RP.A.3 Step 2

Translate 50% more

They attempted 50% more two-point shots than three-point shots.
"50% more than y" means y plus half of y, which is 1.5 times y.
So x equals 1.5y.

$$x = y + 0.5y = 1.5y$$

💡 "50% more" is the original amount plus half of it again.

#8 Analyze the Units 7.RP.A.3 Step 3

Count the points

Half of the x two-point tries go in, each worth 2 points, giving 0.5x times 2 = x points.
Of the y three-point tries, 40% go in, each worth 3 points, giving 0.4y times 3 = 1.2y points.
Together they equal 54.

$$\underbrace{(0.5x)\cdot 2}_{x} + \underbrace{(0.4y)\cdot 3}_{1.2y} = 54 \;\Rightarrow\; x + 1.2y = 54$$

💡 Made shots times point value, added up, must hit the score.

#4 Introduce a Variable 7.EE.B.4 Step 4

Substitute and solve

Replace x with 1.5y in the points equation.
Then 1.5y + 1.2y = 2.7y, and 2.7y = 54 gives y = 20.
So they attempted 20 three-point shots, which is answer (C).

$$1.5y + 1.2y = 2.7y = 54 \;\Rightarrow\; y = \frac{54}{2.7} = 20$$

💡 One substitution turns two unknowns into one solvable equation.

[1] #4 6.EE.B.6 Let x be the number of two-point shots attempted and y be the number of three-po

[2] #13 7.RP.A.3 They attempted 50% more two-point shots than three-point shots. "50% more than y

[3] #8 7.RP.A.3 Half of the x two-point tries go in, each worth 2 points, giving 0.5x times 2 =

[4] #4 7.EE.B.4 Replace x with 1.5y in the points equation. Then 1.5y + 1.2y = 2.7y, and 2.7y =

Review

Reasonableness: With y = 20, x = 1.5(20) = 30. Points: made two-pointers give 0.5(30)(2) = 30, made three-pointers give 0.4(20)(3) = 24, and 30 + 24 = 54. This matches the total, and 20 is one of the listed choices.

Alternative: Test the choices directly. For y = 20, x = 30 gives 30 + 24 = 54, a hit. Smaller choices like y = 10 give 15 + 12 = 27, too low, so the point total pins y down to 20.

CCSS standards used (min grade 7)

6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Assigning letters x and y to the two unknown shot counts.)
7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems (Reading "50% more" as 1.5 times and turning the make-percentages into points.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Building and solving 2.7y = 54 after substitution.)

⭐ Name each unknown with a letter, turn every sentence into an equation, then swap one variable in for the other so only one is left to solve.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Name the two counts

Translate 50% more

Count the points

Substitute and solve

Review

CCSS standards used (min grade 7)

More like this