AMC 8 · 2012 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

ratio-proportionratemulti-digit-arithmetic easier-related-problemdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

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Problem

Rachelle uses 3 pounds of meat for 8 hamburgers. At the same rate, how many pounds of meat does she need for 24 hamburgers?

Pick an answer.

(A)

(B)

$6\dfrac{2}{3}$

(C)

$7\dfrac{1}{2}$

(D)

(E)

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The numbers 8 and 24 are friendly: 24 is exactly 3 batches of 8. Tool #9 (Easier Related Problem) says — instead of fighting a proportion, recognize the smaller version we already solved (3 lb feeds 8 burgers) and copy it three times. Tool #8 (Analyze the Units) keeps the bookkeeping honest: the rate is "pounds per hamburger", so multiplying that rate by a number of hamburgers gives pounds — exactly what the question asks for.

1STEP 1

The big batch of 24 is exactly 3 of the small 8-burger batches (24 ÷ 8).

24 ÷ 8 = 3

2STEP 2

Same recipe, so the meat scales by that same factor of 3, giving 9 pounds.

meat = 3 × 3 lb = 9 lb

3STEP 3

The rate $\frac{3}{8}$ lb per burger times 24 burgers cancels 'burgers' and leaves 9 pounds.

(3 lb)/(8 burgers) × 24 burgers =

\frac{3 × 24}{8}

lb =

\frac{72}{8}

lb = 9 lb → (E)

Answer

Common sense: 3 family-sized batches at 3 pounds each is 9 pounds total — that matches. The answer is bigger than 3 lb (we need more burgers, so we need more meat) and a whole number (because 24 is a whole multiple of 8). Answer choices like 6

\frac{2}{3}

or 7

\frac{1}{2}

would only appear if 24 weren't divisible by 8, so they can be ruled out by structure alone.

💡Key takeaway

When the new batch is just a whole-number multiple of the old batch, you don't need fancy proportions — multiply both sides by the same factor.

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