Answer :
Let's solve this step-by-step:
1. Define the Given Information:
- The probability of picking a red marble is [tex]\(\frac{2}{5}\)[/tex].
- The number of yellow marbles is [tex]\(9\)[/tex].
- The probability of picking a blue marble is three times the probability of picking a yellow marble.
2. Determine the Number of Blue Marbles:
- Let the total number of marbles in the bag be [tex]\(n\)[/tex].
- Let the number of blue marbles be [tex]\(B\)[/tex].
- The probability of picking a blue marble is [tex]\(\frac{B}{n}\)[/tex].
- The probability of picking a yellow marble is [tex]\(\frac{9}{n}\)[/tex].
- We know the probability of picking a blue marble is three times that of picking a yellow marble, so:
[tex]\[ \frac{B}{n} = 3 \times \frac{9}{n} \][/tex]
[tex]\[ B = 3 \times 9 = 27 \][/tex]
Therefore, there are [tex]\(27\)[/tex] blue marbles.
3. Set Up the Equation for the Total Number of Marbles:
- Let the number of red marbles be [tex]\(R\)[/tex].
- The probability of picking a red marble is [tex]\(\frac{R}{n}\)[/tex], which is given as [tex]\(\frac{2}{5}\)[/tex].
- Thus, we have:
[tex]\[ \frac{R}{n} = \frac{2}{5} \][/tex]
Therefore, [tex]\(R = \frac{2}{5} \times n\)[/tex].
4. Write the Total Number of Marbles in Terms of [tex]\(n\)[/tex]:
- The total number of marbles [tex]\(n\)[/tex] is the sum of the red marbles, blue marbles, and yellow marbles:
[tex]\[ n = R + B + 9 \][/tex]
Substitute [tex]\(R\)[/tex] and [tex]\(B\)[/tex] with their respective expressions:
[tex]\[ n = \frac{2}{5}n + 27 + 9 \][/tex]
Simplify:
[tex]\[ n = \frac{2}{5}n + 36 \][/tex]
5. Solve for [tex]\(n\)[/tex]:
- Subtract [tex]\(\frac{2}{5}n\)[/tex] from both sides to isolate [tex]\(n\)[/tex]:
[tex]\[ n - \frac{2}{5}n = 36 \][/tex]
Factor out [tex]\(n\)[/tex] on the left side:
[tex]\[ \left(1 - \frac{2}{5}\right)n = 36 \][/tex]
Simplify the expression within the parentheses:
[tex]\[ \left(\frac{5}{5} - \frac{2}{5}\right)n = 36 \][/tex]
[tex]\[ \left(\frac{3}{5}\right)n = 36 \][/tex]
- Solve for [tex]\(n\)[/tex] by multiplying both sides by [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[ n = 36 \times \frac{5}{3} \][/tex]
[tex]\[ n = 60 \][/tex]
6. Conclusion:
- The total number of marbles in the bag is [tex]\(60\)[/tex].
To summarize:
- There are [tex]\(27\)[/tex] blue marbles.
- There are [tex]\(9\)[/tex] yellow marbles.
- The total number of marbles in the bag is [tex]\(60\)[/tex].
1. Define the Given Information:
- The probability of picking a red marble is [tex]\(\frac{2}{5}\)[/tex].
- The number of yellow marbles is [tex]\(9\)[/tex].
- The probability of picking a blue marble is three times the probability of picking a yellow marble.
2. Determine the Number of Blue Marbles:
- Let the total number of marbles in the bag be [tex]\(n\)[/tex].
- Let the number of blue marbles be [tex]\(B\)[/tex].
- The probability of picking a blue marble is [tex]\(\frac{B}{n}\)[/tex].
- The probability of picking a yellow marble is [tex]\(\frac{9}{n}\)[/tex].
- We know the probability of picking a blue marble is three times that of picking a yellow marble, so:
[tex]\[ \frac{B}{n} = 3 \times \frac{9}{n} \][/tex]
[tex]\[ B = 3 \times 9 = 27 \][/tex]
Therefore, there are [tex]\(27\)[/tex] blue marbles.
3. Set Up the Equation for the Total Number of Marbles:
- Let the number of red marbles be [tex]\(R\)[/tex].
- The probability of picking a red marble is [tex]\(\frac{R}{n}\)[/tex], which is given as [tex]\(\frac{2}{5}\)[/tex].
- Thus, we have:
[tex]\[ \frac{R}{n} = \frac{2}{5} \][/tex]
Therefore, [tex]\(R = \frac{2}{5} \times n\)[/tex].
4. Write the Total Number of Marbles in Terms of [tex]\(n\)[/tex]:
- The total number of marbles [tex]\(n\)[/tex] is the sum of the red marbles, blue marbles, and yellow marbles:
[tex]\[ n = R + B + 9 \][/tex]
Substitute [tex]\(R\)[/tex] and [tex]\(B\)[/tex] with their respective expressions:
[tex]\[ n = \frac{2}{5}n + 27 + 9 \][/tex]
Simplify:
[tex]\[ n = \frac{2}{5}n + 36 \][/tex]
5. Solve for [tex]\(n\)[/tex]:
- Subtract [tex]\(\frac{2}{5}n\)[/tex] from both sides to isolate [tex]\(n\)[/tex]:
[tex]\[ n - \frac{2}{5}n = 36 \][/tex]
Factor out [tex]\(n\)[/tex] on the left side:
[tex]\[ \left(1 - \frac{2}{5}\right)n = 36 \][/tex]
Simplify the expression within the parentheses:
[tex]\[ \left(\frac{5}{5} - \frac{2}{5}\right)n = 36 \][/tex]
[tex]\[ \left(\frac{3}{5}\right)n = 36 \][/tex]
- Solve for [tex]\(n\)[/tex] by multiplying both sides by [tex]\(\frac{5}{3}\)[/tex]:
[tex]\[ n = 36 \times \frac{5}{3} \][/tex]
[tex]\[ n = 60 \][/tex]
6. Conclusion:
- The total number of marbles in the bag is [tex]\(60\)[/tex].
To summarize:
- There are [tex]\(27\)[/tex] blue marbles.
- There are [tex]\(9\)[/tex] yellow marbles.
- The total number of marbles in the bag is [tex]\(60\)[/tex].
