To solve the problem step-by-step, follow these steps:
Step 1: Determine the total ratios and units for each color.
Given:
- The ratio of red to blue marbles is [tex]\(1:2\)[/tex].
- The ratio of red to yellow marbles is [tex]\(2:1\)[/tex].
First, find a common scale for these ratios. Let's set the number of red units as [tex]\(R\)[/tex].
- Since the ratio of red to blue is [tex]\(1:2\)[/tex], the number of blue units will be [tex]\(2R\)[/tex].
- Since the ratio of red to yellow is [tex]\(2:1\)[/tex], the number of yellow units will be [tex]\(\frac{1}{2}R\)[/tex].
Now, to have a common ground, let's use the least common multiple (LCM) to match both given ratios. Solving for the common factors, we will set [tex]\(R\)[/tex] to be [tex]\(2\)[/tex]. Thus:
- Red units: [tex]\(2\)[/tex]
- Blue units: [tex]\(2 \times 2 = 4\)[/tex]
- Yellow units: [tex]\(\frac{1}{2} \times 2 = 1\)[/tex]
The total number of units in the bag is:
[tex]\[ R + 2R + \frac{R}{2} \][/tex]
[tex]\[ = 2 + 4 + 1 \][/tex]
[tex]\[ = 7 \text{ units} \][/tex]
Step 2: Calculate the number of marbles of each color.
Given that the total number of marbles is [tex]\(77\)[/tex], we can determine the number of marbles for each color by dividing the total number of marbles proportionally to the unit representation:
\- Total number of red marbles:
[tex]\[ \frac{77 \times 2}{7} = 22 \][/tex]
\- Total number of blue marbles:
[tex]\[ \frac{77 \times 4}{7} = 44 \][/tex]
\- Total number of yellow marbles:
[tex]\[ \frac{77 \times 1}{7} = 11 \][/tex]
Step 3: Remove two red marbles and recalculate.
After removing 2 red marbles:
[tex]\[ 22 - 2 = 20 \text{ red marbles remaining} \][/tex]
So, the new total number of marbles is:
[tex]\[ 20 (\text{red}) + 44 (\text{blue}) + 11 (\text{yellow}) \][/tex]
[tex]\[ = 75 \text{ marbles} \][/tex]
Step 4: Calculate the new probability of choosing a red marble.
The probability of choosing a red marble from the remaining 75 marbles is:
[tex]\[ P(\text{red}) = \frac{\text{number of red marbles}}{\text{total marbles}} \][/tex]
[tex]\[ P(\text{red}) = \frac{20}{75} \][/tex]
To simplify the fraction:
[tex]\[ \frac{20}{75} = \frac{20 \div 5}{75 \div 5} = \frac{4}{15} \][/tex]
Thus, the new probability of choosing a red marble is:
[tex]\[ \boxed{\frac{4}{15}} \][/tex]
Therefore, the correct answer is C. [tex]\(\frac{4}{15}\)[/tex].
