Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #16
Grade 5 rate-ratioPick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks for one fraction, but to find it we need two separate quantities: how much orange juice is in the container, and how much liquid is in the container in total. Tool #7 (Break into Subproblems) splits the work into those two parallel computations. Tool #4 (Introduce a Variable) is a light support — we treat each pitcher's capacity as the same 600 mL so the two juice amounts can be added directly. Once the two subtotals are in hand, the final fraction is one division and a reduction.
Find the first pitcher's juice
First pitcher: of 600 mL is 200 mL of juice.
Grade 5 "fraction of a whole" — split 600 into 3 equal parts and take one part.
5.NF.B.4Identify SubproblemsFind the second pitcher's juice
Second pitcher: of 600 mL is 240 mL of juice.
Split 600 into 5 equal parts of 120 mL each, then take two of them.
5.NF.B.4Identify SubproblemsAdd the two juice amounts
Add both pitchers' juice: 200 + 240 = 440 mL in the container.
Combining contents = whole-number addition.
4.NBT.B.4Identify SubproblemsFind the total volume
Each pitcher is topped off, so the container holds 600 + 600 = 1200 mL of liquid.
The water fills whatever space the juice did not, so each pitcher gives a full 600 mL.
4.NBT.B.4Identify SubproblemsForm and simplify the fraction
Juice over total: reduces to → (C).
Grade 5 "fraction as division": 440 parts juice out of 1200 parts mixture, reduced to lowest terms by dividing by 10 then by 4.
The share of the combined mixture that is orange juice is found by dividing all the orange juice in the container by all the liquid in the container.
▸ Why?
The juice's share compares the juice to the whole mixture it sits in, and because that mixture is exactly the juice together with the rest of the liquid — no gaps and no overlaps — the juice is a genuine part of the whole, so its share of the whole is that part over the whole.
▸ Why?
All the orange juice in the container is the juice from the first pitcher plus the juice from the second, since pouring both together creates no juice and spills none.
▸ Why?
The container's juice splits with no gaps or overlaps into the part that came from each pitcher, so those two parts add back to the container's total juice.
▸ Why?
All the liquid in the container is 600 mL plus 600 mL, because each pitcher was topped off with water up to its full 600 mL before it was poured in.
▸ Why?
Adding water fills the empty space in a pitcher, so the juice already there and the added water together reach the pitcher's full 600 mL capacity.
Solve two small questions first — how much juice, how much liquid in all — then put one over the other and simplify.
- Find the first pitcher's juice
- Find the second pitcher's juice
- Add the two juice amounts
- Find the total volume
- Form and simplify the fraction
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