AMC 10 · 2002 · #17
Grade 6 rate-ratioPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The process has two pours, so Tool #7 (Identify Subproblems) says handle them one at a time and just keep a running tally of how many ounces of coffee and how many ounces of cream sit in each cup. Tool #8 (Analyze the Units) keeps that tally honest: every number is an amount in ounces, and the final answer is (ounces of cream in Cup 1) divided by (total ounces in Cup 1). The one idea that makes the second pour easy is that a stirred cup is uniform, so pouring back half of it pours back exactly half of its coffee and half of its cream — the ratio is preserved. Tool #4 (Introduce a Variable) is only lightly needed: instead of unknown letters we track two concrete quantities, coffee and cream, through each step.
Pour half the coffee across
Half of Cup 1's 4 ounces moves over: Cup 1 keeps 2 ounces of coffee, and Cup 2 becomes 2 coffee plus 4 cream, 6 ounces in all.
Taking half of a 4-ounce amount just gives 2 ounces, and that coffee simply joins the cream already in the other cup.
5.NF.B.4Analyze The UnitsSee Cup 2 as a fixed mixture
Stirring makes Cup 2 uniform at 2 coffee to 4 cream, so every scoop of it is 1/3 coffee and 2/3 cream.
A well-stirred drink tastes the same in every sip, so any portion of it has coffee and cream in that same 1:2 mix.
A well-stirred drink tastes the same in every sip, so any portion holds the ingredients in the same mix.
▸ Why?
The mixture is uniform, so any share of the volume carries that same share of each ingredient.
▸ Why?
Half of a cup is one of two equal shares, so pouring it back pours back half of each ingredient.
Pour half of Cup 2 back
Half of Cup 2 is 3 ounces, splitting as 1 coffee to 2 cream, so Cup 1 now holds 3 ounces of coffee and 2 ounces of cream.
Because the cup is uniform, pouring back half of it pours back half the coffee and half the cream, so the 3 ounces break into 1 and 2.
6.RP.A.3Identify SubproblemsTake the cream fraction
Cup 1 holds 3 coffee plus 2 cream, 5 ounces in all, so the cream fraction is 2/5 — choice (D).
A fraction of a cup is just the part you care about (cream) divided by everything in the cup.
6.RP.A.3Analyze The UnitsKeep a running count of coffee and cream ounces: a stirred cup is uniform, so pouring back half of it returns half of each, leaving Cup 1 with 2 ounces cream out of 5, which is 2/5.
- Pour half the coffee across
- See Cup 2 as a fixed mixture
- Pour half of Cup 2 back
- Take the cream fraction