Competition · AMC preparation · step 4 of 4
AMC 8 · 2002 · #24
Grade 6 rate-ratioPick an answer.
AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The count of fruit is never specified, which is Tool #11's tell: the percent must be invariant under that count, so any convenient count gives the right answer. Tool #9 (Solve an Easier Problem) lets us pick the friendliest count — 6 of each fruit, the LCM of 3 and 2 — so both juice totals are whole numbers and the percent is a one-line fraction. No algebra needed.
Find each juice rate
Per fruit: one pear gives oz of juice, one orange gives 4 oz (8 oz from 2).
Dividing total juice by number of fruit is the Grade 6 unit-rate move.
6.RP.A.2Solve An Easier Related ProblemPick an equal fruit count
Pick a friendly equal count — 6 of each, the LCM of 3 and 2 — so both juice totals stay whole.
The Grade 6 LCM picks the smallest common count that clears both denominators.
The pear-juice share of the blend is the same for every equal count of pears and oranges, so any convenient count gives the true percent.
▸ Why?
Using n pears and n oranges makes the pear juice n times one pear's juice and the whole blend n times one pear-and-orange pair's juice, so the part and the whole both carry the same factor n.
▸ Why?
n same-size pears pour out n copies of one pear's juice, and n same-size oranges pour out n copies of one orange's juice, so the single count n multiplies each juice amount.
▸ Why?
The share is the pear part over the total, and when both are one base amount times n, that n divides out because n over n is one and multiplying by one changes nothing.
▸ Why?
A count divided by itself is one, since dividing by n undoes the multiplying by n that made the n copies.
▸ Why?
Multiplying the base share by one leaves its value untouched, so the factor n never moves the percent.
Compute both juice totals
At 6 each: 6 pears give 16 oz pear juice (2×8), 6 oranges give 24 oz orange juice (3×8).
Scaling a known 3-pear batch by 2 and a 2-orange batch by 3 — Grade 6 ratio reasoning.
6.RP.A.3Solve An Easier Related ProblemTurn pear juice into a percent
Pear share = 16 over the 40 oz total, times 100 — that gives 40%.
Part divided by whole, times 100, is the Grade 6 "find a percent" recipe.
6.RP.A.3Solve An Easier Related ProblemEach orange out-juices each pear, so pears should be the smaller share. Pick 6 of each fruit to get whole-number ounces — 16 oz pear vs 24 oz orange — and the pear share is = 40%, answer (B).
- Find each juice rate
- Pick an equal fruit count
- Compute both juice totals
- Turn pear juice into a percent
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