2008 AMC 10A Problem 12

Try concrete numbers first to see how the percents work.

1. Suppose there are $4$ blue marbles. "$25\%$ more red than blue" means red is $4$ plus a quarter of $4$, so $5$ red.
2. "$60\%$ more green than red" means green is $5$ plus $60\%$ of $5$, which is $5 + 3 = 8$ green.
3. Total: $4 + 5 + 8 = 17$ marbles, and $17 \div 5 = 3.4$. So the total is $3.4$ times the number of red marbles.

Now the same thing with $r$ red marbles.

4. Red is $1.25$ times blue, so blue is red divided by $1.25$:
$$\text{blue} = \frac{r}{1.25} = \frac{4r}{5} = 0.8r.$$
(In the example: $5 \div 1.25 = 4$. Blue is $20\%$ fewer than red, not $25\%$ fewer.)
5. Green is $60\%$ more than red: $\text{green} = 1.6r$.
6. Add them up:
$$r + 0.8r + 1.6r = 3.4r.$$

The answer is $\boxed{\textbf{(C)}\ 3.4r}$.

"$25\%$ more than blue" measures the $25\%$ from the blue count, so red $= 1.25 \times$ blue. To go back from red to blue you must divide by $1.25$; subtracting $25\%$ of red is a different, wrong step. Trying a small example, like $4$ blue and $5$ red, makes the difference easy to see.

Let $b$ and $g$ be the numbers of blue and green marbles.

"$25\%$ more red than blue" means $r = 1.25b$, so
$$b = \frac{r}{1.25} = \frac{4r}{5} = 0.8r.$$

"$60\%$ more green than red" means $g = 1.6r$.

The total is
$$r + b + g = r + 0.8r + 1.6r = 3.4r.$$

The answer is $\boxed{\textbf{(C)}\ 3.4r}$.