AMC 8 · 2001 · #13

Grade 6 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

fraction-arithmeticratio-proportionmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Short solution 💡 2 insights

Problem

Richelle's class of 36 students has 12 favoring chocolate pie, 8 apple, and 6 blueberry. The remaining students split evenly between cherry and lemon. On a pie graph of this data, how many degrees represent the cherry slice?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break Into Subproblems

The question chains two clean steps: first decide how many students prefer cherry pie, then convert that count into a slice of the 360° circle. Tool #7 (Break Into Subproblems) keeps the two ideas separate. Subproblem (a) is pure arithmetic — subtract the three known counts from 36, then halve. Subproblem (b) is one ratio move — take the cherry fraction of 36 and multiply by 360°. Keeping them apart avoids the common slip of multiplying by 360° before the halving step.

1STEP 1

Subtract the three known groups from 36 to see how many students are left for cherry and lemon: 10 remain.

36 - 12 - 8 - 6 = 10

2STEP 2

Split the 10 remaining students in half, so cherry gets 5 students.

10 ÷ 2 = 5 cherry students

3STEP 3

Cherry is 5 of the 36 students, so its slice is that fraction of 360°: 50°.

\frac{5}{36}

× 360° = 5 × 10° = 50° → (D)

Answer

Check the whole circle adds to 360°. Each student is worth 10°, so chocolate = 120°, apple = 80°, blueberry = 60°, cherry = 50°, lemon = 50°. Sum: 120 + 80 + 60 + 50 + 50 = 360°. The slices fit the full pie, and cherry equals lemon as required. The trap answers come from skipping the halving (taking 10 students × 10° = 100°, then halving the degrees instead of the students) or from using 36 ÷ 360 instead of

\frac{5}{36}

× 360.

💡Key takeaway

When the class size divides 360° cleanly, each student is worth the same number of degrees on the pie graph. Here 360 ÷ 36 = 10° per student, so the 5 cherry fans take 50° — answer (D).

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