AMC 10 · 2019 · #1
Grade 5 rate-ratioPick an answer.
Tool #9 (Easier Problem): pick a friendly number for the second container's volume (say V₂ = 12, the LCM of 6 and 4) so both fractions give whole-number amounts. Then read V₁ off directly. Tool #8 (Units): water amount poured equals water amount received, so V₁ · 5/6 = V₂ · 3/4 — solving for V₁ / V₂ falls right out. Tool #3 (Eliminate): the first container holds less water at 5/6 full than the second at 3/4 full only if V₁ ＜ V₂, so the ratio is less than 1 — all five choices satisfy this, but the clean fraction match picks (D).
Give the second a handy volume
A handy choice makes the water come out clean.
Pick a round size so the fractions become whole numbers.
5.NF.B.6Solve An Easier Related ProblemWrite the first container's condition
That water was five sixths of the first.
Same water, just measured in the first container's frame.
5.NF.B.6Analyze The UnitsFind the first volume
Working back gives the first volume.
Divide by 5/6 is the same as multiplying by 6/5.
Dividing by a fraction is the same as multiplying by it turned over.
▸ Why?
A fraction and its flip multiply back to one, so the flip is exactly what undoes it.
▸ Why?
A fraction counts equal shares of one whole, so undoing it hands the whole back.
Take the ratio
Dividing gives nine tenths.
Simplify the ratio and match it to the answer choices.
5.NF.A.1Eliminate PossibilitiesThis AMC 12 problem only needs Grade 5 fraction multiplication and division you already know — pick the second jar to be 12 units, then the water is 9 units, the first jar is 10.8 units, and the ratio is 9/10.