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Answer :
To solve the problem, let's work through the steps using the information given:
1. Understanding the Problem:
- Initially, there are `r` red counters and `g` green counters.
- The probability of picking a green counter initially is [tex]\(\frac{3}{7}\)[/tex].
2. Setting Up the First Equation:
- The probability of picking a green counter is calculated by dividing the number of green counters by the total number of counters.
- Thus, from the first situation:
[tex]\[
\frac{g}{r + g} = \frac{3}{7}
\][/tex]
- This is our first equation.
3. After Adding Counters:
- We then add 2 more red counters and 3 more green counters to the bag.
- Now, the total number of red counters is [tex]\(r + 2\)[/tex] and green counters is [tex]\(g + 3\)[/tex].
- The probability of picking a green counter becomes [tex]\(\frac{6}{13}\)[/tex].
4. Setting Up the Second Equation:
- Again, the probability of selecting a green counter is the number of green counters divided by the total number of counters:
[tex]\[
\frac{g + 3}{(r + 2) + (g + 3)} = \frac{6}{13}
\][/tex]
- Simplifying the total counters in the denominator gives [tex]\(r + g + 5\)[/tex].
- So, the second equation is:
[tex]\[
\frac{g + 3}{r + g + 5} = \frac{6}{13}
\][/tex]
5. Solving the Equations:
- We solve the two equations simultaneously to find the values of [tex]\(r\)[/tex] and [tex]\(g\)[/tex].
6. Solution:
- From solving these equations, we find that the number of red counters originally was 12, and the number of green counters was 9.
Through this methodical approach, we determine the original numbers of red and green counters in the bag were 12 and 9, respectively. This step-by-step solution enables us to understand how these probabilities relate to the numbers of counters.
1. Understanding the Problem:
- Initially, there are `r` red counters and `g` green counters.
- The probability of picking a green counter initially is [tex]\(\frac{3}{7}\)[/tex].
2. Setting Up the First Equation:
- The probability of picking a green counter is calculated by dividing the number of green counters by the total number of counters.
- Thus, from the first situation:
[tex]\[
\frac{g}{r + g} = \frac{3}{7}
\][/tex]
- This is our first equation.
3. After Adding Counters:
- We then add 2 more red counters and 3 more green counters to the bag.
- Now, the total number of red counters is [tex]\(r + 2\)[/tex] and green counters is [tex]\(g + 3\)[/tex].
- The probability of picking a green counter becomes [tex]\(\frac{6}{13}\)[/tex].
4. Setting Up the Second Equation:
- Again, the probability of selecting a green counter is the number of green counters divided by the total number of counters:
[tex]\[
\frac{g + 3}{(r + 2) + (g + 3)} = \frac{6}{13}
\][/tex]
- Simplifying the total counters in the denominator gives [tex]\(r + g + 5\)[/tex].
- So, the second equation is:
[tex]\[
\frac{g + 3}{r + g + 5} = \frac{6}{13}
\][/tex]
5. Solving the Equations:
- We solve the two equations simultaneously to find the values of [tex]\(r\)[/tex] and [tex]\(g\)[/tex].
6. Solution:
- From solving these equations, we find that the number of red counters originally was 12, and the number of green counters was 9.
Through this methodical approach, we determine the original numbers of red and green counters in the bag were 12 and 9, respectively. This step-by-step solution enables us to understand how these probabilities relate to the numbers of counters.
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