AMC 8 · 1999 · #18

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

percentageratemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: percentagemulti-digit-arithmeticrate

📏 Medium solution 💡 3 insights

Problem

Of the 108 AMC8 students, attendance will be down 25% because of a concert, so only the remaining 75% will show up. Each student eats an average of 2 cookies. A recipe makes a pan of 15 cookies, and only full recipes can be baked. How many recipes must Walter and Gretel make?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The wording stacks four small questions on top of each other — attendance, total cookies, raw recipe count, then round up. Tool #7 (Identify Subproblems) is the cleanest fit: peel the problem apart into a chain of one-step calculations, finish each, then feed it into the next. No algebra needed; each sub-question is a single elementary operation (percent of a number, multiplication, division, ceiling). The only trap is the final "full recipes only" line, which is just a real-world constraint reminding us to round up, not down.

1STEP 1

A quarter stay home, so 75% of the 108 show up: that's 81 students.

Attendees = 0.75 × 108 = 81 students

2STEP 2

Each of the 81 eats 2 cookies on average, so the party needs 162 cookies.

Cookies needed = 81 × 2 = 162

3STEP 3

One pan holds 15 cookies, so 162 ÷ 15 = 10.8 pans — not a whole number.

\frac{162}{15}

= 10.8 recipes

4STEP 4

10 pans make only 150 cookies — short of 162 — so round up to 11 pans, which give 165.

10 × 15 = 150 ＜ 162, 11 × 15 = 165 ≥ 162 → (E) 11

Answer

Quick sanity pass. Attendance dropped by a quarter, so 108 - 27 = 81 — agrees with 0.75 × 108 = 81. Total cookies 81 × 2 = 162 is between 150 (10 recipes) and 165 (11 recipes), so the answer must be 11. Choice (D) 10 would leave 162 - 150 = 12 students short a cookie; choice (E) 11 leaves 165 - 162 = 3 cookies over — exactly the small surplus you'd expect when forced to round up. None of the smaller choices (6, 8, 9) come anywhere near 162 cookies, so they are clearly too few.

💡Key takeaway

Chain the questions: 75% × 108 = 81 attendees, 81 × 2 = 162 cookies, 162 ÷ 15 = 10.8 recipes — and since you cannot bake 0.8 of a pan, round up to 11, answer (E).

More like this