Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #18
Grade 6 rate-ratioPick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find how many students come
A quarter stay home, so 75% of the 108 show up: that's 81 students.
Grade 6 percent-of-a-number: 25% off means 75% left, and 75% of 108 is three-quarters of 108, namely 81.
6.RP.A.3Identify SubproblemsFind the cookies needed
Each of the 81 eats 2 cookies on average, so the party needs 162 cookies.
A Grade 6 unit-rate calculation: 2 cookies per student times 81 students gives 162 cookies total.
6.RP.A.3Identify SubproblemsDivide by the recipe size
One pan holds 15 cookies, so 162 ÷ 15 = 10.8 pans — not a whole number.
Another unit-rate step, this time the other direction: total cookies divided by cookies-per-recipe gives recipes. The .8 tells us 10 recipes is not enough.
6.RP.A.3Identify SubproblemsRound up to whole recipes
10 pans make only 150 cookies — short of 162 — so round up to 11 pans, which give 165.
Grade 4 "interpret remainders": when leftover cookies are still needed, the answer rounds up, not down. Ten pans leave the party twelve cookies short, so eleven it is.
Because only whole recipes may be baked, the fractional recipe count from the division must be rounded up to the next whole number of recipes, not kept as a fraction and not rounded down.
▸ Why?
The cookies produced come only in whole blocks of fifteen, one block per recipe, so the reachable totals jump straight from 150 for ten recipes to 165 for one more, with no amount in between.
▸ Why?
One recipe is one equal group of fifteen cookies, so making r recipes is r equal groups of fifteen, exactly r times fifteen cookies and never a partial amount.
▸ Why?
Splitting the 162 cookies needed into fifteen-cookie blocks fits ten whole blocks and leaves 12 cookies over, so ten recipes make only 150 and fall short of what the party needs.
▸ Why?
Dividing the 162 needed by the 15 in each block is just undoing the equal grouping, and it reports how many full blocks fit, ten here, with a remainder of 12 still unfilled.
▸ Why?
Those 12 leftover cookies are still owed to the party and cannot be dropped, and since a partial recipe cannot be baked, covering them forces one more whole recipe on top of the ten.
▸ Why?
The 162 needed is the whole made of the 150 from ten recipes plus the 12 still required, and every part of that whole has to be produced, so the missing 12 cannot simply be ignored.
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).
- Find how many students come
- Find the cookies needed
- Divide by the recipe size
- Round up to whole recipes
A parent dashboard for the family lives at sensimlab.com.
More like this
Same problem type. Tags show what each one shares with this problem.
- AMC 8 2001 #13 Gr 6Similar levelSame sub-type: Linear Rate AccumulationSame techniqueIdentify Subproblems
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the rem…
- AMC 8 2001 #5 Gr 6Similar levelSame sub-type: Linear Rate AccumulationSame techniqueIdentify Subproblems
On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. Th…
- AMC 8 2002 #7 Gr 6Similar level
Same sub-type: Linear Rate AccumulationSame techniqueIdentify SubproblemsThe students in Mrs. Sawyer's class were asked to do a taste test of five kinds of candy. Each student chose one kind of…