AMC 10 · 2008 · #4

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

ratio-proportionratefraction-multiplication dimensional-analysiseasier-related-problem ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 1 insight

Problem

Every banana has the same value and every orange has the same value. We are told that

\frac{2}{3}

of 10 bananas is worth as much as 8 oranges. At that same exchange rate, find how many oranges are worth as much as

\frac{1}{2}

of 5 bananas.

Pick an answer.

(A)

$\ 2$

(B)

$\ \frac{5}{2}$

(C)

$\ 3$

(D)

$\ \frac{7}{2}$

(E)

$\ 4$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

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 "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.

1STEP 1

Simplify the given amount of bananas

" $\frac{2}{3}$ of 10 bananas" is just $\frac{2}{3}\times 10=\frac{20}{3}$ , so the given fact reads $\frac{20}{3}$ bananas = 8 oranges.

2/3 × 10=20/3 bananas = 8 oranges

2STEP 2

Find the value of one banana in oranges

Share the 8 oranges evenly over those bananas: $8\div\frac{20}{3}=8\times\frac{3}{20}=\frac{6}{5}$ , so one banana is worth $\frac{6}{5}$ oranges.

1 banana=8÷20/3=8×3/20=6/5 oranges

3STEP 3

Simplify the target amount of bananas

The asked side simplifies the same way: $\frac{1}{2}\times 5=\frac{5}{2}$ , so we need the orange value of $\frac{5}{2}$ bananas.

1/2 × 5=5/2 bananas

4STEP 4

Scale by the rate to get the answer

Multiply by the rate: $\frac{5}{2}\times\frac{6}{5}$ — the 5s cancel and 6 halves to 3, so the answer is 3 oranges, choice (C).

5/2×6/5=6/2=3 oranges → (C)

Answer

Check with whole-number scaling to avoid fraction slips. The given says 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 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 5/2 usually means forgetting to divide by the 2/3.

💡Key takeaway

Find what one banana is worth in oranges first; after that, any pile of bananas becomes oranges with a single multiply.

Simplify the given amount of bananas
Find the value of one banana in oranges
Simplify the target amount of bananas
Scale by the rate to get the answer

Simplify the given amount of bananas

Find the value of one banana in oranges

Simplify the target amount of bananas

Scale by the rate to get the answer

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