Answer :
To solve this problem, we need to find the original number of red counters ([tex]\( r \)[/tex]) and green counters ([tex]\( g \)[/tex]) in the bag.
### Step 1: Set Up the Equations
1. Initial Probability Equation:
The probability that a counter drawn is green is given initially by:
[tex]\[
\frac{g}{r + g} = \frac{4}{9}
\][/tex]
This equation tells us the relationship between the green and red counters initially.
2. Modified Probability Equation:
After adding more counters, the new total number of red counters is [tex]\( r + 4 \)[/tex] and the number of green counters is [tex]\( g + 2 \)[/tex].
The new probability that a counter drawn is green is:
[tex]\[
\frac{g + 2}{(r + 4) + (g + 2)} = \frac{10}{23}
\][/tex]
### Step 2: Solve the Equations
Now, we have a system of equations to solve:
1. From the initial condition:
[tex]\[
9g = 4(r + g)
\][/tex]
2. From the modified condition:
[tex]\[
23(g + 2) = 10((r + 4) + (g + 2))
\][/tex]
Let's simplify each step.
For equation (1):
- Multiply out the equation:
[tex]\[
9g = 4r + 4g
\][/tex]
- Rearrange to find:
[tex]\[
5g = 4r \quad \Rightarrow \quad g = \frac{4}{5}r
\][/tex]
For equation (2):
- Expand and simplify the equation:
[tex]\[
23g + 46 = 10r + 60 + 10g + 20
\][/tex]
[tex]\[
23g + 46 = 10r + 10g + 80
\][/tex]
- Rearrange to:
[tex]\[
13g = 10r + 34
\][/tex]
### Step 3: Solve Simultaneously
Using the expressions obtained:
Substitute [tex]\( g = \frac{4}{5}r \)[/tex] into the adjusted equation:
- Replace [tex]\( g \)[/tex]:
[tex]\[
13\left(\frac{4}{5}r\right) = 10r + 34
\][/tex]
[tex]\[
\frac{52}{5}r = 10r + 34
\][/tex]
- Clear the fraction by multiplying all terms by 5:
[tex]\[
52r = 50r + 170
\][/tex]
- Simplify:
[tex]\[
2r = 170 \quad \Rightarrow \quad r = 85
\][/tex]
Now find [tex]\( g \)[/tex] using [tex]\( g = \frac{4}{5}r \)[/tex]:
- Substitute:
[tex]\[
g = \frac{4}{5} \times 35 = 28
\][/tex]
### Final Result
The original number of red counters is 35, and the original number of green counters is 28.
### Step 1: Set Up the Equations
1. Initial Probability Equation:
The probability that a counter drawn is green is given initially by:
[tex]\[
\frac{g}{r + g} = \frac{4}{9}
\][/tex]
This equation tells us the relationship between the green and red counters initially.
2. Modified Probability Equation:
After adding more counters, the new total number of red counters is [tex]\( r + 4 \)[/tex] and the number of green counters is [tex]\( g + 2 \)[/tex].
The new probability that a counter drawn is green is:
[tex]\[
\frac{g + 2}{(r + 4) + (g + 2)} = \frac{10}{23}
\][/tex]
### Step 2: Solve the Equations
Now, we have a system of equations to solve:
1. From the initial condition:
[tex]\[
9g = 4(r + g)
\][/tex]
2. From the modified condition:
[tex]\[
23(g + 2) = 10((r + 4) + (g + 2))
\][/tex]
Let's simplify each step.
For equation (1):
- Multiply out the equation:
[tex]\[
9g = 4r + 4g
\][/tex]
- Rearrange to find:
[tex]\[
5g = 4r \quad \Rightarrow \quad g = \frac{4}{5}r
\][/tex]
For equation (2):
- Expand and simplify the equation:
[tex]\[
23g + 46 = 10r + 60 + 10g + 20
\][/tex]
[tex]\[
23g + 46 = 10r + 10g + 80
\][/tex]
- Rearrange to:
[tex]\[
13g = 10r + 34
\][/tex]
### Step 3: Solve Simultaneously
Using the expressions obtained:
Substitute [tex]\( g = \frac{4}{5}r \)[/tex] into the adjusted equation:
- Replace [tex]\( g \)[/tex]:
[tex]\[
13\left(\frac{4}{5}r\right) = 10r + 34
\][/tex]
[tex]\[
\frac{52}{5}r = 10r + 34
\][/tex]
- Clear the fraction by multiplying all terms by 5:
[tex]\[
52r = 50r + 170
\][/tex]
- Simplify:
[tex]\[
2r = 170 \quad \Rightarrow \quad r = 85
\][/tex]
Now find [tex]\( g \)[/tex] using [tex]\( g = \frac{4}{5}r \)[/tex]:
- Substitute:
[tex]\[
g = \frac{4}{5} \times 35 = 28
\][/tex]
### Final Result
The original number of red counters is 35, and the original number of green counters is 28.
