Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #17
Grade 6 rate-ratiopatternPick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question has five separate candidates, so Tool #7 (Identify Subproblems) splits the work cleanly: compute one percent increase per candidate, then compare. Tool #6 (Guess and Check) is the right second tool because we are literally testing each choice against the formula and keeping the smallest. No algebra is needed — just the percent-increase formula five times and a head-to-head comparison.
Write the percent-increase formula
Set up the percent-increase formula, always dividing by the older (smaller-numbered) question's prize.
Treating the older value as the "whole" (the base of 100%) is the Grade 6 percent-change convention.
6.RP.A.3Identify SubproblemsApply it to all five pairs
Plug each pair into the formula and reduce to get 100%, 50%, about 66.7%, about 95.3%, and 100%.
For (D), the values are in thousands but the units cancel in the ratio, so (125-64)/64 uses the same arithmetic as cents or dollars.
Applying percent increase = (V_new - V_old)/V_old to the five listed prize pairs gives 100%, 50%, ≈ 66.7%, ≈ 95.3%, and 100%.
▸ Why?
A percent increase compares the jump in prize to the older prize, so for each pair you find the jump and then divide it by that older value.
▸ Why?
The jump is the newer prize minus the older prize.
▸ Why?
The newer prize is the older prize with the added dollars stacked on top, so it splits into the original part and the added part.
▸ Why?
Subtracting the older prize then peels the original part back off and leaves exactly the added dollars, because subtraction undoes that addition.
▸ Why?
Dividing that jump by the older prize and reading the result as a count out of a hundred is exactly what the percent increase reports, since the older prize is the whole the percent is measured against.
Pick the smallest percent
Line up the five percents and pick the smallest: 50% wins.
Ordering five rational numbers and picking the smallest is a Grade 6 number-line comparison.
6.NS.C.7Identify SubproblemsWhen a prize jumps by the same dollars, the percent change shrinks as the starting value grows — so $200 → $300 is a 50% jump, the smallest of the five pairs.
- Write the percent-increase formula
- Apply it to all five pairs
- Pick the smallest percent
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