AMC 8 · 2007 · #17

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentageratio-proportionfraction-arithmetic identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Short solution 💡 2 insights

Problem

A 30-liter paint mixture is 25% red tint, 30% yellow tint, and 45% water. Five more liters of pure yellow tint are stirred in. What percent of the new mixture is yellow tint?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The trap in this problem is reasoning with the percent 30% directly — when you add liquid to one part, both the part and the whole change, so the percent does not just go up by some neat number. Tool #7 (Identify Subproblems) handles this by splitting the work into three independent steps: (a) convert the 30% to actual liters of yellow, (b) update the liters of yellow and the total liters after the pour, and (c) convert back to a percent. Each step is one arithmetic move. Tool #1 (Draw a Diagram) supports this with a part-whole bar that makes the "yellow piece grows, total bar grows" picture explicit. We deliberately avoid Tool #13 (Algebra) because no equation is needed — just three lines of arithmetic on amounts.

1STEP 1

Convert the starting percent into liters: 30% of the 30-liter mixture gives 9 liters of yellow.

yellow_old = 30% × 30 = 0.30 × 30 = 9 liters

2STEP 2

Pour in 5 more liters of yellow: the yellow rises to 14 liters while red and water stay put.

yellow_new = 9 + 5 = 14 liters

3STEP 3

The 5 poured-in liters push the total mixture from 30 up to 35 liters.

total_new = 30 + 5 = 35 liters

4STEP 4

Divide the new yellow by the new total: $\frac{14}{35}$ simplifies to $\frac{2}{5}$ , which is 40%.

yellow_new/total_new =

\frac{14}{35}

\frac{2}{5}

= 40% → (C)

Answer

Sanity-check the size of the answer. The yellow share started at 30% and we added more yellow, so the new percent must be greater than 30% — this rules out (A) 25. But the addition is only 5 liters compared to a total of 35, so the jump cannot be huge — 50% would mean half the mixture is yellow, which would need 17.5 liters of yellow, far more than the 14 we have. The actual jump from 30% to 40% is exactly 10 percentage points, which fits the modest size of the pour. As a final check: the red tint and water keep their original amounts (25% × 30 = 7.5 and 45% × 30 = 13.5), and 7.5 + 13.5 + 14 = 35 matches the new total.

💡Key takeaway

When a mixture problem asks about percent, work in actual amounts first — turn 30% into 9 liters, do the addition, then turn the answer back into a percent. The numbers do the rest.

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