AMC 8 · 2010 · #11

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionlinear-equations-one-var ratio-proportionidentify-subproblems ↑ Prerequisites: ratio-proportionlinear-equations-one-var

📏 Medium solution 💡 3 insights

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Problem

Two trees stand side by side. The taller tree's top is 16 feet above the shorter tree's top, and the two heights are in the ratio 3:4 (short to tall). In feet, how tall is the taller tree?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

112

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The ratio 3:4 already gives us a tiny version of the problem — two "trees" of heights 3 and 4, differing by 1. Tool #9 (Easier Related Problem) says: solve the small case first, then scale it up until the difference matches the real 16 feet. That avoids algebra entirely. Tool #3 (Eliminate Possibilities) is the multiple-choice safety net: the taller tree must be divisible by 4 (so the shorter side $\frac{3}{4}$ of it is a whole number), which already cuts the choices down.

1STEP 1

Shrink it: pretend the two trees are 3 and 4 feet tall — that fits the 3:4 ratio, and the gap is 4 - 3 = 1 foot.

small case: heights = 3, 4 → gap = 4 - 3 = 1

2STEP 2

The real gap is 16 feet and the tiny gap is 1 foot, so the scale factor is 16.

scale factor =

\frac{16}{1}

= 16

3STEP 3

Scale both: the shorter tree is 3 × 16 = 48 feet and the taller tree is 4 × 16 = 64 feet, still 3:4 with a 16-foot gap.

3 × 16 = 48, 4 × 16 = 64, 64 - 48 = 16 ✓

4STEP 4

Check the choices: a quarter of the taller tree must equal the 16-foot gap, so only 64 — choice (B) — fits.

\frac{1}{4}

h = 16 → h = 64 → (B)

Answer

Heights of 48 ft and 64 ft are realistic for medium-sized trees, and 64 - 48 = 16 ft matches the given gap exactly. The ratio 48 : 64 simplifies to 3 : 4 by dividing both by 16, so both problem conditions are satisfied. The answer (B) is the only choice that pairs with a whole-number partner

\frac{3}{4}

· 64 = 48 differing by exactly 16.

💡Key takeaway

This AMC 8 problem only needs Grade 6 ratio reasoning — scale a small 3{:}4 pair up until the gap matches!

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