AMC 8 · 2022 · #12

Grade 7 probabilitycounting

Archetype Small-Domain Constrained Search

probability-basicperfect-squaressystematic-enumeration systematic-enumerationcasework ↑ Prerequisites: probability-basicperfect-squares

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

The arrows on the two spinners shown below are spun. Let the number $N$ equal $10$ times the number on Spinner $\text{A}$ , added to the number on Spinner $\text{B}$ . What is the probability that $N$ is a perfect square number?

Pick an answer.

(A)

$\dfrac{1}{16}$

(B)

$\dfrac{1}{8}$

(C)

$\dfrac{1}{4}$

(D)

$\dfrac{3}{8}$

(E)

$\dfrac{1}{2}$

View mode:

Toolkit + CCSS Solution

Understand

Restated: Spinner A is split into $4$ equal regions labeled $5, 6, 7, 8$, and Spinner B is split into $4$ equal regions labeled $1, 2, 3, 4$. Spin both, then form the two-digit number $N = 10 \times (\text{Spinner A}) + (\text{Spinner B})$. Find the probability that $N$ is a perfect square.

Givens: Spinner A outcomes: $\{5, 6, 7, 8\}$, each with probability $\tfrac{1}{4}$; Spinner B outcomes: $\{1, 2, 3, 4\}$, each with probability $\tfrac{1}{4}$; $N = 10 \times A + B$, so Spinner A is the tens digit and Spinner B is the ones digit; Both spinners are spun independently; Answer choices: (A) $\tfrac{1}{16}$, (B) $\tfrac{1}{8}$, (C) $\tfrac{1}{4}$, (D) $\tfrac{3}{8}$, (E) $\tfrac{1}{2}$

Unknowns: $P(N \text{ is a perfect square})$

Understand

Plan

Primary tool: #2 Make a Systematic List

Secondary: #7 Identify Subproblems, #6 Guess and Check, #1 Draw a Diagram

The sample space has only $4 \times 4 = 16$ outcomes — small enough to enumerate exhaustively, so Tool #2 (Systematic List) is the perfect fit. Tool #7 (Identify Subproblems) splits the work into two clean pieces: (a) count the total outcomes ($16$), and (b) count how many produce a perfect square. For (b), instead of squaring every two-digit value, use Tool #6 (Guess and Check) on the small set of perfect squares actually in range — $8^2 = 64$ and $9^2 = 81$ are the only candidates because $7^2 = 49 < 51$ and $10^2 = 100 > 84$. Then check whether those squares' digits really match what the spinners can produce.

Execute — Answer: B

#7 Identify Subproblems 3.OA.A.1 Step 1

Count the size of the sample space.
Each spinner has $4$ equally likely results, and the spins are independent, so the total number of ordered outcomes $(A, B)$ is $4 \times 4 = 16$.
Each of these $16$ pairs is equally likely.

$$\text{Total outcomes} = 4 \times 4 = 16$$

💡 Multiplying "$4$ choices on A" by "$4$ choices on B" is the Grade 3 "groups of equal size" view of multiplication.

#1 Draw a Diagram 1.NBT.B.2 Step 2

Find the range of $N$.
Because the tens digit is $5, 6, 7,$ or $8$ and the ones digit is $1, 2, 3,$ or $4$, the smallest possible $N$ is $51$ and the largest is $84$.
So we only need to check which perfect squares lie between $51$ and $84$.

$$51 \le N \le 84$$

💡 Reading the two-digit number as "tens digit + ones digit" is Grade 1 place value.

#6 Guess and Check 3.OA.C.7 Step 3

List the perfect squares near this range to find the only candidates inside it.
$7^2 = 49$ is just below $51$, $8^2 = 64$ is inside, $9^2 = 81$ is inside, and $10^2 = 100$ is above $84$.
So the only perfect-square values of $N$ in range are $64$ and $81$.

$$7^2 = 49,\ 8^2 = 64,\ 9^2 = 81,\ 10^2 = 100 \;\Rightarrow\; \{64, 81\}$$

💡 Recognizing $64 = 8 \times 8$ and $81 = 9 \times 9$ comes straight from Grade 3 multiplication fluency.

#2 Make a Systematic List 1.NBT.B.2 Step 4

Check that each candidate can actually be produced by the spinners.
For $N = 64$, the tens digit is $6$ (yes, on Spinner A) and the ones digit is $4$ (yes, on Spinner B) — so the pair $(A, B) = (6, 4)$ works.
For $N = 81$, the tens digit is $8$ (on Spinner A) and the ones digit is $1$ (on Spinner B) — so $(A, B) = (8, 1)$ works.
That gives exactly $2$ favorable outcomes out of $16$.

$$\text{Favorable pairs} = \{(6,4),\ (8,1)\} \;\Rightarrow\; 2 \text{ outcomes}$$

💡 Splitting $64$ into "$6$ in the tens place, $4$ in the ones place" is the same Grade 1 place-value move.

#2 Make a Systematic List 7.SP.C.7 Step 5

Compute the probability as favorable divided by total, then reduce the fraction.
Of the $16$ equally likely outcomes, $2$ are favorable, so the probability is $\tfrac{2}{16}$.
Dividing top and bottom by $2$ gives $\tfrac{1}{8}$, which is choice (B).

$$P(N \text{ is a perfect square}) = \dfrac{2}{16} = \dfrac{1}{8} \;\Rightarrow\; \textbf{(B)}$$

💡 Forming a probability as $\dfrac{\text{favorable outcomes}}{\text{all outcomes}}$ for equally likely outcomes is the Grade 7 probability-model recipe.

[1] #7 3.OA.A.1 Count the size of the sample space. Each spinner has $4$ equally likely results,

[2] #1 1.NBT.B.2 Find the range of $N$. Because the tens digit is $5, 6, 7,$ or $8$ and the ones

[3] #6 3.OA.C.7 List the perfect squares near this range to find the only candidates inside it.

[4] #2 1.NBT.B.2 Check that each candidate can actually be produced by the spinners. For $N = 64$

[5] #2 7.SP.C.7 Compute the probability as favorable divided by total, then reduce the fraction.

Review

Reasonableness: Out of $16$ tiny equally likely outcomes, only $2$ are perfect squares — that is a small fraction, so the answer should be small. $\tfrac{1}{8} = 0.125$ is small and matches choice (B). Choices (D) $\tfrac{3}{8}$ and (E) $\tfrac{1}{2}$ would require $6$ or $8$ perfect squares among the $16$ values, but the range $51$–$84$ obviously cannot contain that many squares (squares get sparser as numbers grow). And $\tfrac{1}{16}$ (only $1$ favorable) would miss either $64$ or $81$.

Alternative: Tool #1 (Draw a Diagram): build a $4 \times 4$ grid with rows labeled $A = 5, 6, 7, 8$ and columns labeled $B = 1, 2, 3, 4$. Write the value of $N = 10A + B$ in each of the $16$ cells, then circle the ones that are perfect squares. Two cells get circled ($64$ at row $6$/column $4$, and $81$ at row $8$/column $1$), so the probability is $\tfrac{2}{16} = \tfrac{1}{8}$. The grid makes the count visual and impossible to double-count.

CCSS standards used (min grade 7)

1.NBT.B.2 Understand that the two digits of a two-digit number represent tens and ones (Reading $N = 10A + B$ as "$A$ in the tens place, $B$ in the ones place" — used to find the range $51$–$84$ and to verify $64 = (6)(4)$ and $81 = (8)(1)$ are reachable.)
3.OA.A.1 Interpret products of whole numbers as total number of objects in groups (Counting the sample space as $4 \times 4 = 16$ ordered $(A, B)$ pairs from two independent spinners.)
3.OA.C.7 Fluently multiply and divide within 100 (Recognizing the perfect squares near the range: $7^2 = 49$, $8^2 = 64$, $9^2 = 81$, $10^2 = 100$ — a Grade 3 multiplication-facts check.)
7.SP.C.7 Develop probability models and use them to find probabilities of events (Defining the probability of "$N$ is a perfect square" as $\dfrac{\text{favorable}}{\text{total}} = \dfrac{2}{16} = \dfrac{1}{8}$ using a uniform probability model on $16$ equally likely outcomes.)

⭐ This AMC 8 problem only needs the Grade 7 "favorable over total" probability model you already know — the perfect-square hunt is just Grade 3 multiplication facts!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 7)

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