AMC 10 · 2014 · #6
Grade 8 rate-ratioPick an answer.
Every amount in the story is described relative to the regular size, so Tool #4 (Introduce a Variable) names that size r and Tool #13 (Convert to Algebra) turns each sentence into an expression in r. The efficient move is Tool #16 (Change Focus): instead of tracking two glasses, notice that the handover does not change the total, so the pair drinks 5/2r no matter what, and 'they drank the same amount' means each drank 5/4r. Tool #11 (Work Backwards) then reads off how big the handover has to be for that to happen, while Tool #7 (Identify Subproblems) computes how big the handover actually is, in two pieces: a third of Ann's leftover, then 2 more ounces. Two descriptions of one quantity give one linear equation. Tool #6 (Guess and Check) closes the argument by putting r back into the story and replaying it, which is what shows the answer is possible and not just forced.
Name the regular size
One letter names both servings.
A percent increase scales the original, so 50% more is the same as multiplying by 3/2.
6.RP.A.3Introduce A VariableLook at the total, not the glasses
Each person drinks exactly half the total.
Pouring between two cups never changes how much lemonade is on the table.
Pouring between two glasses never changes how much lemonade is on the table.
▸ Why?
What one glass loses the other gains, so the total stays exactly where it was.
▸ Why?
The total is the two glasses added together, so tracking it is simpler than tracking either one.
How big must the handover be
That fixes how big the handover must be.
A transfer closes a gap twice as fast as its own size, so it has to be half the gap.
6.EE.A.2Work BackwardsHow big the handover actually is
The story says how big it actually is.
The 3/4 matters only because it fixes how much is left for the 'one third' to be taken from.
5.NF.B.6Identify SubproblemsOne equation, one regular size
Setting them equal gives a serving of 16.
Two honest descriptions of one quantity are exactly the equation you need.
8.EE.C.7Convert To AlgebraReplay the story, then add up
Replaying the story gives 40, choice (D).
An equation tells you what the answer must be; replaying the story shows the story can really happen.
6.EE.A.2Guess And CheckPassing lemonade changes who drinks it but not how much there is, so the handover just has to close exactly half the gap between the two drinks.
- Name the regular size
- Look at the total, not the glasses
- How big must the handover be
- How big the handover actually is
- One equation, one regular size
- Replay the story, then add up