AMC 10 · 2020 · #1
Grade 6 arithmeticPick an answer.
The pie is eaten in two separate rounds, so split the question into two small ones (Tool #7): how much is left after Carlos, then how much of that survives Maria. Tool #9 makes the percents concrete by treating the pie as 100 equal parts, so every quantity is a whole number of parts instead of a symbol. Tool #16 handles the last round by tracking what Maria leaves rather than what she takes: one third taken means two thirds stay, which reaches the answer in one multiplication and doubles as a check on the subtraction.
Cut the pie into 100 parts
Find what Carlos left behind.
Percent just means parts out of 100, so counting leftover pie is counting leftover parts.
6.RP.A.3Solve An Easier Related ProblemMaria's third is of the remainder
Maria takes a third of that leftover.
A fraction always needs a base, and here the base is the leftover pie, not the original pie.
A fraction always needs a base, and here the base is the leftover pie, not the original one.
▸ Why?
A fraction is a count of equal shares of whatever whole it is applied to.
▸ Why?
A percent is a count out of a hundred of that same whole, so the base decides the meaning.
Two thirds of the remainder stays
Two thirds of the leftover stays, so 20 percent.
If one third leaves, two thirds stay, so you can multiply once instead of taking away.
5.NF.B.4Change Focus Count The ComplementWhen a problem says a fraction of the remainder, that fraction is measured against what is left, not against the whole thing.
- Cut the pie into 100 parts
- Maria's third is of the remainder
- Two thirds of the remainder stays