AMC 8 · 2002 · #24
Grade 6 rate-ratioProblem
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
Pick an answer.
AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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?
Givens: $3$ pears yield $8$ oz of pear juice, so the rate is $\tfrac{8}{3}$ oz per pear; $2$ oranges yield $8$ oz of orange juice, so the rate is $\tfrac{8}{2} = 4$ oz per orange; The blend uses the same number of pears as oranges; Answer choices: (A) $30$, (B) $40$, (C) $50$, (D) $60$, (E) $70$
Unknowns: The percent of the blend that is pear juice
Understand
Restated: 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?
Givens: $3$ pears yield $8$ oz of pear juice, so the rate is $\tfrac{8}{3}$ oz per pear; $2$ oranges yield $8$ oz of orange juice, so the rate is $\tfrac{8}{2} = 4$ oz per orange; The blend uses the same number of pears as oranges; Answer choices: (A) $30$, (B) $40$, (C) $50$, (D) $60$, (E) $70$
Plan
Primary tool: #9 Solve an Easier Problem
Secondary: #11 Find an Invariant
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.
Execute — Answer: B
6.RP.A.2 Step 1 - Find the per-fruit juice rates.
- The juicer makes $8$ oz from $3$ pears, so each pear gives $\tfrac{8}{3}$ oz.
- It makes $8$ oz from $2$ oranges, so each orange gives $\tfrac{8}{2} = 4$ oz.
💡 Dividing total juice by number of fruit is the Grade 6 unit-rate move.
6.NS.B.4 Step 2 - Pick a friendly equal count.
- Choose $n = 6$, the LCM of $3$ and $2$, so neither rate produces a fraction.
- The percent will not depend on $n$, so we just want clean arithmetic.
💡 The Grade 6 LCM picks the smallest common count that clears both denominators.
6.RP.A.3 Step 3 - Compute each juice total at $n = 6$.
- From $6$ pears: $6 \div 3 = 2$ batches of $8$ oz, so $16$ oz of pear juice.
- From $6$ oranges: $6 \div 2 = 3$ batches of $8$ oz, so $24$ oz of orange juice.
💡 Scaling a known $3$-pear batch by $2$ and a $2$-orange batch by $3$ — Grade 6 ratio reasoning.
6.RP.A.3 Step 4 Form the pear-juice percent of the blend.
💡 Part divided by whole, times $100$, is the Grade 6 "find a percent" recipe.
6.RP.A.2 Find the per-fruit juice rates. The juicer makes $8$ oz from $3$ pears, so each 6.NS.B.4 Pick a friendly equal count. Choose $n = 6$, the LCM of $3$ and $2$, so neither 6.RP.A.3 Compute each juice total at $n = 6$. From $6$ pears: $6 \div 3 = 2$ batches of $ 6.RP.A.3 Form the pear-juice percent of the blend. Review
Reasonableness: Check invariance with a different count. Take $n = 1$: pear juice $= \tfrac{8}{3}$ oz, orange juice $= 4$ oz, so the pear share is $\dfrac{8/3}{8/3 + 4} = \dfrac{8/3}{20/3} = \dfrac{8}{20} = 40\%$ — same answer, confirming the count did not matter. The sign also fits intuition: each orange gives more juice ($4$ oz) than each pear ($\tfrac{8}{3} \approx 2.67$ oz), so pears should make up less than half the blend, and $40\%$ is comfortably under $50\%$.
Alternative: Tool #4 (Introduce a Variable): let $n$ be the equal number of each fruit. Pear juice $= \tfrac{8n}{3}$, orange juice $= 4n$. Then $\dfrac{8n/3}{8n/3 + 4n} = \dfrac{8/3}{8/3 + 4} = \dfrac{8}{8 + 12} = \dfrac{8}{20} = 40\%$. The $n$ cancels — that algebraic cancellation is the reason a single convenient count was enough in the main path.
CCSS standards used (min grade 6)
6.RP.A.2Understand the concept of a unit rate $a/b$ associated with a ratio $a:b$ with $b \neq 0$, and use rate language in the context of a ratio relationship (Converting $8$ oz from $3$ pears into the unit rate $\tfrac{8}{3}$ oz/pear, and $8$ oz from $2$ oranges into $4$ oz/orange.)6.NS.B.4Find the greatest common factor of two whole numbers less than or equal to $100$ and the least common multiple of two whole numbers less than or equal to $12$ (Choosing $n = \operatorname{lcm}(3, 2) = 6$ as the friendliest equal count so both juice totals are whole numbers.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, including finding a percent of a quantity as a rate per $100$ (Scaling the per-batch yields to $6$ pears and $6$ oranges and computing $\tfrac{16}{40} \times 100 = 40\%$.)
⭐ 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 $\tfrac{16}{40} = 40\%$, answer (B).
⭐ 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 $\tfrac{16}{40} = 40\%$, answer (B).
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