AMC 10 · 2004 · #14

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

fraction-arithmeticratio-proportion convert-to-algebra ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

A bag holds only red and blue marbles, with more blue than red. First, red marbles are added until blue marbles are

\frac{1}{3}

of the bag. Next, yellow marbles are added until blue marbles are

\frac{1}{5}

of the bag. Finally, the number of blue marbles is doubled. Find what fraction of the marbles are blue at the end.

Pick an answer.

(A)

$\frac{1}{5}$

(B)

$\frac{1}{4}$

(C)

$\frac{1}{3}$

(D)

$\frac{2}{5}$

(E)

$\frac{1}{2}$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

The last action is doubling the blue marbles, so the only snapshot that decides the answer is the bag right before that: blue marbles are 1/5 of it. Everything earlier — the 1/3 stage, the starting counts — is a distraction that never touches the final fraction, which is why Tool #9 (Solve an Easier Related Problem) lets us throw it away. The winning move is Tool #16 (Change Focus / Count the Complement): stop chasing the blue count and watch the non-blue marbles, because doubling blue leaves them untouched. Tool #4 (Introduce a Variable) turns the 1/5 into concrete counts so the doubling is easy to track.

1STEP 1

Keep only the last snapshot

Only the bag just before the doubling matters, and there blue is $\frac{1}{5}$ of the total — the earlier $\frac{1}{3}$ stage can be ignored.

just before doubling: blue = 1/5 of the bag

2STEP 2

Split the bag into 5 equal parts

Picture the bag as 5 equal parts of $b$ marbles: $b$ blue and $4b$ non-blue, total $5b$ .

blue=b, non-blue=4b, total=5b

3STEP 3

Double the blue, leave the rest alone

Doubling makes blue $2b$ ; the non-blue $4b$ is untouched, so the new total is $6b$ .

blue=2b, non-blue=4b, total=2b+4b=6b

4STEP 4

Recount the blue fraction

Blue is now $2b$ out of $6b$ , so the blue share is $\frac{2b}{6b}=\frac{1}{3}$ — choice (C).

blue/total=2b/6b=1/3 (C)

Answer

1/3

Put in real numbers with

b=10

: start with 10 blue and 40 non-blue, a total of 50, and indeed

\frac{10}{50}=\frac{1}{5}

is blue. Doubling gives 20 blue and still 40 non-blue, total 60, and

\frac{20}{60}=

\frac{1}{3}

— choice (C). Notice the trap: doubling a

\frac{1}{5}

share does not give

\frac{2}{5}

(choice (D)), because the total grows too. The blue share rises from

\frac{1}{5}

\frac{1}{3}

, a modest increase, which fits the fact that blue was still a minority after doubling.

💡Key takeaway

Doubling the blue marbles does not double their fraction — the non-blue pile stays put, so 1 part blue out of 5 becomes 2 parts out of 6, which is $\frac{1}{3}$ .

Keep only the last snapshot
Split the bag into 5 equal parts
Double the blue, leave the rest alone
Recount the blue fraction

Keep only the last snapshot

Split the bag into 5 equal parts

Double the blue, leave the rest alone

Recount the blue fraction

More like this