Competition · AMC preparation · step 4 of 4
AMC 8 · 2007 · #6
Grade 6 rate-ratioPick an answer.
AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Identify Subproblems) breaks the question into three small steps: (a) find the amount of decrease, (b) form the ratio of decrease to the original price, and (c) convert that ratio to a percent. Each piece is a single arithmetic move. Tool #6 (Guess and Check) handles the word "approximate" — the answer choices are widely spaced (7, 17, 34, 41, 80), so a quick estimate of 34/41 is enough to pick the only choice close to it. We avoid Tool #13 (Algebra) because no variable is needed.
Find the decrease
The price fell from 41 cents to 7 cents, so subtract to get the drop: 34 cents.
"How much did it go down?" is just subtraction of the two prices.
4.NBT.B.4Identify SubproblemsSet up the percent ratio
Percent decrease compares the drop to the original 1985 price, so the ratio is .
Percent change always uses the starting amount as the whole, so 41 goes on the bottom.
The percent decrease is the 34-cent drop measured against the original 41-cent price, giving the ratio 34/41.
▸ Why?
The top of the fraction is the 34-cent drop — the amount the price lost when it fell from 41 cents to 7 cents.
▸ Why?
The original 41 cents splits with no overlap into the 7 cents still paid and the amount lost, so the amount lost is exactly 41 - 7 = 34.
▸ Why?
The bottom of the fraction is the original 41 cents, so the ratio measures what portion of the starting price disappeared.
▸ Why?
The starting 41-cent price is the whole because it is the complete amount the drop came out of, while the 7-cent 2005 price is only the leftover part, not the whole.
▸ Why?
Dividing the 34-cent drop by 41 gives that portion, because the portion lost is the number that times 41 returns 34, and dividing undoes that multiplying.
Estimate the fraction as a percent
Since 34 is nearly 41, the ratio sits just under 1 — about 83%.
Dividing by something close to the numerator gives a value close to (but under) 100% — a useful sanity check.
6.RP.A.3Identify SubproblemsPick the closest choice
The choices are spread far apart, so 83% rounds to the nearest one, 80% — choice (E).
When choices are spread out, a rough estimate is enough — no exact decimal needed.
6.RP.A.3Guess And CheckPercent decrease is just "how much it dropped" divided by "what it started at" — one subtraction and one division get you the answer.
- Find the decrease
- Set up the percent ratio
- Estimate the fraction as a percent
- Pick the closest choice
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