山姆有一袋红色、蓝色和绿色的弹珠。红色弹珠数量与蓝色弹珠数量的比为 2 : 1。蓝色弹珠数量与绿色弹珠数量的比为 5 : 3。

1. Identify the two given ratios and the common term

Red : Blue = $2 : 1$
Blue : Green = $5 : 3$
The common term is Blue.

2. Make the common term equal in both ratios to form a combined ratio

Multiply the first ratio by 5 to match the Blue units in the second ratio:
Red : Blue = $(2 \times 5) : (1 \times 5) = 10 : 5$
Now we can combine them:
Red : Blue : Green = $10 : 5 : 3$

3. Calculate the total number of units and the fraction of green marbles

Total units = $10 + 5 + 3 = 18$
Fraction of green marbles = $\frac{\text{Green units}}{\text{Total units}} = \frac{3}{18} = \frac{1}{6}$

1. Establish the initial ratio of red to blue marbles

From the problem, the initial ratio of red to blue marbles is $2 : 1$. Let the initial number of red marbles be $2u$ and blue marbles be $1u$.

2. Set up equations based on the removal of marbles

Let $n$ be the number of rounds. In each round, 4 red and 3 blue marbles are removed.
Final red marbles: $2u - 4n = 18$
Final blue marbles: $1u - 3n = 1$

3. Solve the simultaneous equations for u and n

From the second equation, $u = 3n + 1$.
Substitute into the first equation:
$2(3n + 1) - 4n = 18$
$6n + 2 - 4n = 18$
$2n = 16$
$n = 8$
Then, $u = 3(8) + 1 = 25$.

4. Calculate the initial number of red marbles

Initial red marbles = $2u = 2(25) = 50$.