AMC 10 · 2008 · #4
Grade 6 rate-ratioSuppose that 32 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as 21 of 5 bananas?
Pick an answer.
AMC 10 2008 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: Bananas and oranges each have a fixed value. We are told that $\tfrac{2}{3}$ of $10$ bananas is worth the same as $8$ oranges. Using that exchange rate, find how many oranges are worth as much as $\tfrac{1}{2}$ of $5$ bananas.
Givens: $\tfrac{2}{3}$ of $10$ bananas is worth the same as $8$ oranges; Every banana has the same value, and every orange has the same value; Answer choices: (A) $2$, (B) $\frac{5}{2}$, (C) $3$, (D) $\frac{7}{2}$, (E) $4$
Unknowns: The number of oranges worth as much as $\tfrac{1}{2}$ of $5$ bananas
Understand
Restated: Bananas and oranges each have a fixed value. We are told that $\tfrac{2}{3}$ of $10$ bananas is worth the same as $8$ oranges. Using that exchange rate, find how many oranges are worth as much as $\tfrac{1}{2}$ of $5$ bananas.
Givens: $\tfrac{2}{3}$ of $10$ bananas is worth the same as $8$ oranges; Every banana has the same value, and every orange has the same value; Answer choices: (A) $2$, (B) $\frac{5}{2}$, (C) $3$, (D) $\frac{7}{2}$, (E) $4$
Plan
Primary tool: #8 Analyze the Units
Secondary: #9 Solve an Easier Related Problem, #4 Introduce a Variable
The whole problem is one exchange rate: oranges per banana. Tool #8 (Analyze the Units) says to lock onto that rate — find how many oranges a single banana is worth, and every question about bananas becomes a multiplication. Tool #9 (Solve an Easier Related Problem) is why we shrink the messy "$\tfrac{2}{3}$ of $10$ bananas" down to the value of just one banana first. Tool #4 (Introduce a Variable) backs this up: calling one banana $b$ oranges keeps the balance equation honest.
Execute — Answer: C
5.NF.B.4 Step 1 Simplify the given amount of bananas
- First turn the words into a plain count of bananas.
- "$\tfrac{2}{3}$ of $10$ bananas" means $\tfrac{2}{3}\times 10=\tfrac{20}{3}$ bananas.
- So the given fact is simply: $\tfrac{20}{3}$ bananas are worth $8$ oranges.
💡 "$\tfrac{2}{3}$ of $10$" is just a multiplication, so replace the phrase with the single number it stands for.
6.RP.A.2 Step 2 Find the value of one banana in oranges
- If $\tfrac{20}{3}$ bananas are worth $8$ oranges, then one banana is worth that $8$ shared among $\tfrac{20}{3}$ bananas: $8\div\tfrac{20}{3}=8\times\tfrac{3}{20}=\tfrac{24}{20}=\tfrac{6}{5}$.
- So one banana is worth $\tfrac{6}{5}$ of an orange.
- This single rate carries the whole problem.
💡 Split the total orange value evenly across the bananas to see what a single banana is worth.
5.NF.B.4 Step 3 Simplify the target amount of bananas
- Now handle what the question asks about.
- "$\tfrac{1}{2}$ of $5$ bananas" means $\tfrac{1}{2}\times 5=\tfrac{5}{2}$ bananas.
- So we need the orange value of $\tfrac{5}{2}$ bananas.
💡 Same move as before: turn the phrase into one clean banana count.
6.RP.A.3 Step 4 Scale by the rate to get the answer
- Each banana is worth $\tfrac{6}{5}$ orange, so $\tfrac{5}{2}$ bananas are worth $\tfrac{5}{2}\times\tfrac{6}{5}$ oranges.
- The $5$s cancel, leaving $\tfrac{6}{2}=3$.
- So $\tfrac{1}{2}$ of $5$ bananas is worth $3$ oranges, which is choice (C).
💡 Once you know oranges-per-banana, any banana count converts to oranges by one multiplication.
5.NF.B.4 First turn the words into a plain count of bananas. "$\tfrac{2}{3}$ of $10$ bana 6.RP.A.2 If $\tfrac{20}{3}$ bananas are worth $8$ oranges, then one banana is worth that 5.NF.B.4 Now handle what the question asks about. "$\tfrac{1}{2}$ of $5$ bananas" means $ 6.RP.A.3 Each banana is worth $\tfrac{6}{5}$ orange, so $\tfrac{5}{2}$ bananas are worth Review
Reasonableness: Check with whole-number scaling to avoid fraction slips. The given says $\tfrac{20}{3}$ bananas $=8$ oranges; multiply both sides by $3$ to get $20$ bananas $=24$ oranges, so $5$ bananas $=6$ oranges. Half of that is $\tfrac{1}{2}$ of $5$ bananas $=3$ oranges, matching (C). The value $3$ sits sensibly in the middle of the choices $2$ through $4$, and picking a smaller value like $\tfrac{5}{2}$ usually means forgetting to divide by the $\tfrac{2}{3}$.
Alternative: Introduce a variable: let one banana be worth $b$ oranges. The given fact is $\tfrac{20}{3}b=8$, so $b=\tfrac{6}{5}$. The target is $\tfrac{5}{2}b=\tfrac{5}{2}\cdot\tfrac{6}{5}=3$, giving (C) directly from the balance equation.
CCSS standards used (min grade 6)
5.NF.B.4Multiply a fraction by a whole number or fraction (Turning "$\tfrac{2}{3}$ of $10$ bananas" into $\tfrac{20}{3}$ and "$\tfrac{1}{2}$ of $5$ bananas" into $\tfrac{5}{2}$.)6.RP.A.2Understand the concept of a unit rate and use rate language (Finding the value of a single banana, $\tfrac{6}{5}$ orange per banana, as the unit rate that drives the whole problem.)6.RP.A.3Use ratio and rate reasoning to solve real-world problems (Scaling the oranges-per-banana rate up to $\tfrac{5}{2}$ bananas to get $3$ oranges.)
⭐ Find what one banana is worth in oranges first; after that, any pile of bananas becomes oranges with a single multiply.
⭐ Find what one banana is worth in oranges first; after that, any pile of bananas becomes oranges with a single multiply.
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