AMC 8 · 2022 · #12
Grade 7 probabilitycountingProblem
The arrows on the two spinners shown below are spun. Let the number N equal 10 times the number on Spinner A, added to the number on Spinner B. What is the probability that N is a perfect square number?
Pick an answer.
AMC 8 2022 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.
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
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) $\tfrac{1}{2}$
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
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.
💡 Multiplying "$4$ choices on A" by "$4$ choices on B" is the Grade 3 "groups of equal size" view of multiplication.
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 $84$.
💡 Reading the two-digit number as "tens digit + ones digit" is Grade 1 place value.
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 $81$.
💡 Recognizing $64 = 8 \times 8$ and $81 = 9 \times 9$ comes straight from Grade 3 multiplication fluency.
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 $16$.
💡 Splitting $64$ into "$6$ in the tens place, $4$ in the ones place" is the same Grade 1 place-value move.
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).
💡 Forming a probability as $\dfrac{\text{favorable outcomes}}{\text{all outcomes}}$ for equally likely outcomes is the Grade 7 probability-model recipe.
3.OA.A.1 Count the size of the sample space. Each spinner has $4$ equally likely results, 1.NBT.B.2 Find the range of $N$. Because the tens digit is $5, 6, 7,$ or $8$ and the ones 3.OA.C.7 List the perfect squares near this range to find the only candidates inside it. 1.NBT.B.2 Check that each candidate can actually be produced by the spinners. For $N = 64$ 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.2Understand 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.1Interpret 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.7Fluently 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.7Develop 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!
⭐ 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!
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