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2004 AMC 10B Problem 14

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2004 AMC 10B Problem 14AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P11-15

A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only 13\frac{1}{3} of the marbles in the bag are blue. Then yellow marbles are added to the bag until only 15\frac{1}{5} of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?

  1. A)

    15\frac{1}{5}

  2. B)

    14\frac{1}{4}

  3. C)

    13\frac{1}{3}

  4. D)

    25\frac{2}{5}

  5. E)

    12\frac{1}{2}

Answer

C

Key insight

Only the last snapshot matters: blue is 1/5 of the bag, so blue : other = 1 : 4, and doubling blue makes it 2 : 4, i.e. one third.

Solution

The red and yellow steps only set the stage; what matters is the bag just before the last step, when blue marbles are 15\frac{1}{5} of the total. Write that as xx blue marbles and 4x4x non-blue marbles.

Doubling the blue marbles gives 2x2x blue, while the 4x4x others are untouched. The bag now holds 2x+4x=6x2x + 4x = 6x marbles, of which
2x6x=13\frac{2x}{6x} = \frac{1}{3}
are blue.

The answer is (C) 13\boxed{\textbf{(C)}\ \frac{1}{3}}.

Why this works

A fraction of the whole is a part-to-whole ratio; to see the effect of changing one part, convert to a part-to-part ratio (blue : other =1:4= 1 : 4), change the part, and convert back. The earlier steps are a distraction: multi-stage word problems often only need the final state.

Alternative approach

Use concrete numbers. Start with 22 blue and 11 red; add 33 red so blue is 22 of 66 (one third); add 44 yellow so blue is 22 of 1010 (one fifth); double blue to 44 of 1212, which is 13\frac{1}{3}.

The trap

Doubling the fraction 1/5 to 2/5 (choice (D)), forgetting that the added blue marbles also enlarge the total.

Common mistakes

  • Doubling the fraction 1/5 to 2/5 (choice (D)), forgetting that the added blue marbles also enlarge the total.
  • Getting tangled in the red and yellow stages and trying to track all four colors symbolically when only the final ratio is needed.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer