To determine the total number of marbles in the bag, let's start by defining and working with the given probabilities and known quantities:
1. Probability of Red Marble: The probability of picking a red marble is given as [tex]\(\frac{2}{5}\)[/tex].
2. Number of Yellow Marbles: There are 9 yellow marbles in the bag.
3. Probability of Blue Marble: The probability of picking a blue marble is three times the probability of picking a yellow marble.
Let the total number of marbles in the bag be [tex]\(T\)[/tex].
Step-by-Step Solution:
1. Probability of Picking a Yellow Marble:
Since there are 9 yellow marbles out of a total of [tex]\(T\)[/tex] marbles, the probability of picking a yellow marble is:
[tex]\[
P(Y) = \frac{9}{T}
\][/tex]
2. Probability of Picking a Blue Marble:
Given that the probability of picking a blue marble is three times the probability of picking a yellow marble, we have:
[tex]\[
P(B) = 3 \times P(Y) = 3 \times \frac{9}{T} = \frac{27}{T}
\][/tex]
3. Sum of All Probabilities:
The sum of the probabilities of picking either a red, blue, or yellow marble must equal 1 (since one of these outcomes must happen when a marble is picked). Therefore, we can write:
[tex]\[
P(R) + P(Y) + P(B) = 1
\][/tex]
Substituting the known probabilities:
[tex]\[
\frac{2}{5} + \frac{9}{T} + \frac{27}{T} = 1
\][/tex]
4. Combine Like Terms:
Combine the terms involving [tex]\(T\)[/tex]:
[tex]\[
\frac{2}{5} + \frac{9 + 27}{T} = 1
\][/tex]
Simplify the fraction:
[tex]\[
\frac{2}{5} + \frac{36}{T} = 1
\][/tex]
5. Isolate and Solve for [tex]\(T\)[/tex]:
To isolate [tex]\(T\)[/tex], we need to clear the fraction. First, subtract [tex]\(\frac{2}{5}\)[/tex] from both sides:
[tex]\[
\frac{36}{T} = 1 - \frac{2}{5}
\][/tex]
Simplify the right-hand side:
[tex]\[
1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}
\][/tex]
So we have:
[tex]\[
\frac{36}{T} = \frac{3}{5}
\][/tex]
6. Cross Multiply to Solve for [tex]\(T\)[/tex]:
Cross-multiply to clear the fractions:
[tex]\[
36 \times 5 = 3 \times T
\][/tex]
Simplify:
[tex]\[
180 = 3T
\][/tex]
Divide both sides by 3:
[tex]\[
T = \frac{180}{3} = 60
\][/tex]
Therefore, the total number of marbles in the bag is:
[tex]\[
T = 60
\][/tex]
So, the total number of marbles in the bag is 60.
