Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #11
Grade 6 rate-ratioPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 3/4 of it is a whole number), which already cuts the choices down.
Try a smaller case
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.
Replacing the unknown heights with the ratio numbers themselves is the cleanest "easier problem" — a Grade 6 ratio idea.
6.RP.A.1Solve An Easier Related ProblemFind the scale factor
The real gap is 16 feet and the tiny gap is 1 foot, so the scale factor is 16.
"Multiply both numbers by the same factor" is exactly Grade 4 multiplicative comparison.
4.OA.A.2Solve An Easier Related ProblemScale both tree heights
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.
Scaling a ratio by a common factor preserves it — the heart of Grade 6 ratio reasoning.
Multiplying both numbers of the small 3,4 pair by the same factor 16 gives a taller pair that still stands in the 3:4 ratio and whose gap has grown from 1 to exactly 16 feet.
▸ Why?
Scaling both heights by the same factor 16 leaves their 3:4 comparison unchanged, because a ratio compares the two amounts by division and the shared factor 16 appears on both sides, so it cancels out.
▸ Why?
The shared factor 16/16 is 1, because dividing by 16 undoes multiplying by 16.
▸ Why?
Once that shared factor is seen to be 1, the 3-to-4 comparison multiplied by 1 is just the 3-to-4 comparison again — nothing changes.
▸ Why?
The gap between the two heights is also multiplied by 16, so the small gap of 1 becomes a gap of 16, because both scaled numbers carry the same factor 16 and that shared factor pulls out of their difference: 16 · 4 - 16 · 3 = 16·(4-3) = 16 · 1.
Check against the choices
Check the choices: a quarter of the taller tree must equal the 16-foot gap, so only 64 — choice (B) — fits.
Testing each choice against "1/4 of the taller tree equals the gap" is the Tool #3 elimination move.
6.RP.A.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 ratio reasoning — scale a small 3{:}4 pair up until the gap matches!
- Try a smaller case
- Find the scale factor
- Scale both tree heights
- Check against the choices
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