AMC 10 · 2011 · #12

Grade 7 arithmetic

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

systems-of-equationslinear-equations-one-var convert-to-algebra ↑ Prerequisites: linear-equations-one-var

📏 Medium solution 💡 2 insights

Problem

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2011 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.

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Toolkit + CCSS Solution

Understand

Restated: A team scores with three-point shots, two-point shots, and one-point free throws. The points from two-point shots equal the points from three-point shots. The number of free throws is one more than the number of two-point shots. The whole team scored 61 points. Find how many free throws they made.

Givens: Points scored on two-point shots equal points scored on three-point shots; Number of free throws = number of two-point shots + 1; Each free throw is worth 1 point; Total score is 61 points

Unknowns: The number of free throws made

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #13 Convert to Algebra, #16 Change Focus / Count the Complement

Everything is tied to the number of two-point shots: the free throws are one more than it, and the equal-points rule lets me write the three-point total without ever counting three-point shots. So I name that one number, turn each clue into an expression in dollars of points, and solve a single equation.

Execute — Answer: A

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

Name the number of two-point shots

Let $t$ be the number of two-point shots.
Each is worth 2 points, so the points from two-point shots are $2t$.

$$\text{two-point points} = 2t$$

💡 Pin down the one quantity every other clue depends on, and the rest can be written in terms of it.

#16 Change Focus / Count the Complement 6.EE.B.6 Step 2

Use the equal-points rule for threes

The points from three-point shots equal the points from two-point shots.
So the three-point points are also $2t$.
I never need to count how many three-point shots there were.

$$\text{three-point points} = \text{two-point points} = 2t$$

💡 Counting points instead of shots turns 'equal points' into a copy of a value I already have.

#4 Introduce a Variable 6.EE.B.6 Step 3

Write the free-throw points

The number of free throws is one more than the number of two-point shots, so it is $t + 1$.
Each free throw is worth 1 point, so the free-throw points are also $t + 1$.

$$\text{free-throw points} = t + 1$$

💡 A one-point shot means its point total is the same number as its count.

#13 Convert to Algebra 7.EE.B.4 Step 4

Add the parts to 61

The three point sources add to the total: $2t$ from twos, $2t$ from threes, and $t + 1$ from free throws sum to 61.
Combining like terms gives $5t + 1 = 61$.

$$2t + 2t + (t + 1) = 61 \;\Rightarrow\; 5t + 1 = 61$$

💡 Adding up every scoring channel must land on the known grand total, which gives one equation.

#13 Convert to Algebra 7.EE.B.4 Step 5

Solve and read off the free throws

Subtract 1 from both sides to get $5t = 60$, so $t = 12$.
The number of free throws is $t + 1 = 13$.
That is choice (A).

$$5t = 60 \;\Rightarrow\; t = 12 \;\Rightarrow\; t + 1 = 13$$

💡 Undo the equation step by step to free the variable, then translate back to the thing asked for.

[1] #4 6.EE.B.6 Let $t$ be the number of two-point shots. Each is worth 2 points, so the points

[2] #16 6.EE.B.6 The points from three-point shots equal the points from two-point shots. So the

[3] #4 6.EE.B.6 The number of free throws is one more than the number of two-point shots, so it

[4] #13 7.EE.B.4 The three point sources add to the total: $2t$ from twos, $2t$ from threes, and

[5] #13 7.EE.B.4 Subtract 1 from both sides to get $5t = 60$, so $t = 12$. The number of free thr

Review

Reasonableness: Check with $t = 12$: two-point shots give $24$ points, three-point shots give the same $24$ points (that is $8$ three-point shots, a whole number), and $13$ free throws give $13$ points. The total is $24 + 24 + 13 = 61$, matching the problem, and $13$ is choice (A). The count of free throws being one more than the $12$ two-point shots also checks out.

Alternative: Track only the points from twos as a single block $P$. Threes also give $P$, and free throws give (two-point shots) $+ 1 = \tfrac{P}{2} + 1$. Then $P + P + \tfrac{P}{2} + 1 = 61$, so $\tfrac{5P}{2} = 60$ and $P = 24$; free throws $= \tfrac{24}{2} + 1 = 13$, the same answer.

CCSS standards used (min grade 7)

6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Naming the number of two-point shots as $t$ and writing each scoring source ($2t$, $2t$, $t+1$) as an expression in $t$.)
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities (Building the two-step equation $5t + 1 = 61$ and solving it for $t$ to find the free throws.)

⭐ Anchor everything to one unknown, use 'equal points' to reuse a value you already wrote, then add all the scoring to the total and solve.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Name the number of two-point shots

Use the equal-points rule for threes

Write the free-throw points

Add the parts to 61

Solve and read off the free throws

Review

CCSS standards used (min grade 7)

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