2004 AMC 10B Problem 14

The red and yellow steps just lead up to one moment: right before the last step, blue is $\frac{1}{5}$ of the bag. Start there.

1. Pick a concrete number. Say the bag has $5$ marbles then, with $1$ blue and $4$ not blue.
2. Double the blue marbles: now $2$ blue and still $4$ not blue. The bag has $2 + 4 = 6$ marbles.
3. The fraction that is blue is $\frac{2}{6} = \frac{1}{3}$.
4. Any other size gives the same answer. With $x$ blue and $4x$ others, doubling gives $2x$ blue out of $2x + 4x = 6x$, and $\frac{2x}{6x} = \frac{1}{3}$.

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

A fraction of the whole changes in a tricky way when you change one part, because the whole changes too. It is easier to compare the parts: blue to others is $1$ to $4$, doubling blue makes it $2$ to $4$, and then you turn it back into a fraction of the whole. Also, in a story with many steps, check whether only the last step matters.

Follow the whole story with real numbers. Start with $2$ blue and $1$ red (more blue than red, as the problem says). Add $3$ red: now $2$ blue out of $6$, which is $\frac{1}{3}$. Add $4$ yellow: $2$ blue out of $10$, which is $\frac{1}{5}$. Double blue: $4$ blue out of $12$, which is $\frac{1}{3}$.