AMC 8 · 2002 · #24

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

rateratio-proportionpercentagefraction-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Short solution 💡 2 insights

Problem

A juicer gives 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. Miki blends an equal number of pears and oranges. What percent of the blend is pear juice?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Problem

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.

1STEP 1

Per fruit: one pear gives $\frac{8}{3}$ oz of juice, one orange gives 4 oz (8 oz from 2).

pear rate =

\frac{8}{3}

oz/pear, orange rate = 4 oz/orange

2STEP 2

Pick a friendly equal count — 6 of each, the LCM of 3 and 2 — so both juice totals stay whole.

n = lcm(3, 2) = 6 pears and 6 oranges

3STEP 3

At 6 each: 6 pears give 16 oz pear juice (2×8), 6 oranges give 24 oz orange juice (3×8).

pear juice =

\frac{6}{3}

· 8 = 16 oz, orange juice =

\frac{6}{2}

· 8 = 24 oz

4STEP 4

Pear share = 16 over the 40 oz total, times 100 — that gives 40%.

16/(16 + 24) × 100 =

\frac{16}{40}

× 100 = 40% → (B)

Answer

Check invariance with a different count. Take n = 1: pear juice =

\frac{8}{3}

oz, orange juice = 4 oz, so the pear share is (

\frac{8}{3}

)/(

\frac{8}{3}

+ 4) = (

\frac{8}{3}

)/(

\frac{20}{3}

) =

\frac{8}{20}

= 40% — same answer, confirming the count did not matter. The sign also fits intuition: each orange gives more juice (4 oz) than each pear (

\frac{8}{3}

≈ 2.67 oz), so pears should make up less than half the blend, and 40% is comfortably under 50%.

💡Key takeaway

Each 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 $\frac{16}{40}$ = 40%, answer (B).

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