AMC 10 · 2019 · #1
Grade 7 geometry-2dPick an answer.
This is two small questions stacked, so split them (Tool #7): first get each pizza's area from the radius, then turn the two areas into a percent comparison. Tool #8 keeps the percent honest by asking "percent of what?" — the phrase "larger than the radius-3 pizza" makes the smaller area the base, not the larger one. Tool #9 then strips the problem down: π sits in both areas and cancels, so the messy circle numbers collapse into a plain fraction of whole numbers.
Write each pizza's area
Write each pizza's area.
Area lives in two dimensions, so the radius is squared before π scales it.
7.G.B.4Identify SubproblemsDecide what the percent compares
The percent is against the smaller one.
"Larger than X" always means the growth divided by X, not the new amount divided by X.
Larger than something always means the growth divided by that something, not the new amount divided by it.
▸ Why?
A percent is a count out of a hundred of the thing being compared against, so the base decides the meaning.
▸ Why?
The growth is the difference between the two amounts, which is what the comparison is really about.
Cancel π and divide
The pi cancels.
A ratio of two areas never cares about π, because π scales both of them equally.
6.RP.A.3Solve An Easier Related ProblemConvert to a percent and round
As a percent, rounded, that is 78.
A percent is just the fraction rescaled so that the base counts as 100.
7.NS.A.2Identify Subproblems"Percent larger" means the extra amount divided by what you started with, so subtract first and then divide by the smaller one.
- Write each pizza's area
- Decide what the percent compares
- Cancel π and divide
- Convert to a percent and round