AMC 10 · 2018 · #1
Grade 6 rate-ratioPick an answer.
The question asks "how many," so name that count with a variable (Tool #4) and let the rest of the problem describe it. Tool #8 (Analyze the Units) does the first piece of work: percent means "per hundred," so with exactly 100 balls the percentages convert straight into ball counts, and the whole problem becomes counting instead of percenting. The key structural fact is that removing blue balls leaves the red count frozen, so only the total moves — which turns the 72% condition into a single equation about the new total. Tool #13 (Convert to Algebra) writes that condition down and solves it.
Turn the percents into ball counts
Percentages become actual counts.
With exactly 100 objects, a percent and a count are the same number.
6.RP.A.3Analyze The UnitsOnly the total changes
Red stays put; only the total shrinks.
The red balls never move, so the only thing the percentage can respond to is the shrinking total.
The red balls never move, so the only thing the percentage can respond to is the shrinking total.
▸ Why?
Only blue balls leave, so what is taken from one part is taken from the whole and nowhere else.
▸ Why?
A percentage is a part over the whole, so with the part fixed the whole alone decides it.
Write the 72% condition
The 72 percent condition is one fraction.
A percentage is just a part-over-whole fraction, so the condition is one equation, not a process.
6.RP.A.3Convert To AlgebraFind the final total, then x
Fifty remain, so 50 were taken out.
Same numerator, so the fractions are equal exactly when the denominators are equal — that pins the final total in one line.
6.EE.B.7Convert To AlgebraIf you only take away the other color, the color you care about keeps its count — so a percent question is really a question about what the new total has to be.
- Turn the percents into ball counts
- Only the total changes
- Write the 72% condition
- Find the final total, then x