Answer :
To solve this problem, let's break it down step by step using the information provided in the question:
1. Understand the Initial Situation:
- Let the number of red counters originally be [tex]\( r \)[/tex].
- Let the number of green counters originally be [tex]\( g \)[/tex].
- The total number of counters initially is [tex]\( r + g \)[/tex].
- The probability of drawing a green counter initially is given as [tex]\(\frac{4}{9}\)[/tex].
[tex]\[
\frac{g}{r + g} = \frac{4}{9}
\][/tex]
2. Set up the First Equation:
- Multiply both sides by [tex]\( r + g \)[/tex] to eliminate the fraction:
[tex]\[
9g = 4(r + g)
\][/tex]
- Simplify the equation:
[tex]\[
9g = 4r + 4g
\][/tex]
[tex]\[
5g = 4r
\][/tex]
[tex]\[
r = \frac{5}{4}g
\][/tex]
3. Understand the Situation After Adding Counters:
- After adding 4 red counters and 2 green counters, the new numbers are:
- Red counters: [tex]\( r + 4 \)[/tex]
- Green counters: [tex]\( g + 2 \)[/tex]
- Total counters: [tex]\( r + g + 6 \)[/tex]
- The probability of drawing a green counter in this situation is given as [tex]\(\frac{10}{23}\)[/tex].
[tex]\[
\frac{g + 2}{r + g + 6} = \frac{10}{23}
\][/tex]
4. Set up the Second Equation:
- Multiply both sides by [tex]\( r + g + 6 \)[/tex]:
[tex]\[
23(g + 2) = 10(r + g + 6)
\][/tex]
- Simplify the equation:
[tex]\[
23g + 46 = 10r + 10g + 60
\][/tex]
[tex]\[
13g + 46 = 10r + 60
\][/tex]
[tex]\[
13g = 10r + 14
\][/tex]
5. Solve the System of Equations:
- Now, use the equations [tex]\( r = \frac{5}{4}g \)[/tex] and [tex]\( 13g = 10r + 14 \)[/tex].
- Substitute [tex]\( r = \frac{5}{4}g \)[/tex] into the second equation:
[tex]\[
13g = 10\left(\frac{5}{4}g\right) + 14
\][/tex]
[tex]\[
13g = \frac{50}{4}g + 14
\][/tex]
[tex]\[
13g = 12.5g + 14
\][/tex]
[tex]\[
0.5g = 14
\][/tex]
[tex]\[
g = 28
\][/tex]
- Substitute [tex]\( g = 28 \)[/tex] back into [tex]\( r = \frac{5}{4}g \)[/tex]:
[tex]\[
r = \frac{5}{4} \times 28
\][/tex]
[tex]\[
r = 35
\][/tex]
Thus, the number of red counters originally in the bag is 35, and the number of green counters is 28.
1. Understand the Initial Situation:
- Let the number of red counters originally be [tex]\( r \)[/tex].
- Let the number of green counters originally be [tex]\( g \)[/tex].
- The total number of counters initially is [tex]\( r + g \)[/tex].
- The probability of drawing a green counter initially is given as [tex]\(\frac{4}{9}\)[/tex].
[tex]\[
\frac{g}{r + g} = \frac{4}{9}
\][/tex]
2. Set up the First Equation:
- Multiply both sides by [tex]\( r + g \)[/tex] to eliminate the fraction:
[tex]\[
9g = 4(r + g)
\][/tex]
- Simplify the equation:
[tex]\[
9g = 4r + 4g
\][/tex]
[tex]\[
5g = 4r
\][/tex]
[tex]\[
r = \frac{5}{4}g
\][/tex]
3. Understand the Situation After Adding Counters:
- After adding 4 red counters and 2 green counters, the new numbers are:
- Red counters: [tex]\( r + 4 \)[/tex]
- Green counters: [tex]\( g + 2 \)[/tex]
- Total counters: [tex]\( r + g + 6 \)[/tex]
- The probability of drawing a green counter in this situation is given as [tex]\(\frac{10}{23}\)[/tex].
[tex]\[
\frac{g + 2}{r + g + 6} = \frac{10}{23}
\][/tex]
4. Set up the Second Equation:
- Multiply both sides by [tex]\( r + g + 6 \)[/tex]:
[tex]\[
23(g + 2) = 10(r + g + 6)
\][/tex]
- Simplify the equation:
[tex]\[
23g + 46 = 10r + 10g + 60
\][/tex]
[tex]\[
13g + 46 = 10r + 60
\][/tex]
[tex]\[
13g = 10r + 14
\][/tex]
5. Solve the System of Equations:
- Now, use the equations [tex]\( r = \frac{5}{4}g \)[/tex] and [tex]\( 13g = 10r + 14 \)[/tex].
- Substitute [tex]\( r = \frac{5}{4}g \)[/tex] into the second equation:
[tex]\[
13g = 10\left(\frac{5}{4}g\right) + 14
\][/tex]
[tex]\[
13g = \frac{50}{4}g + 14
\][/tex]
[tex]\[
13g = 12.5g + 14
\][/tex]
[tex]\[
0.5g = 14
\][/tex]
[tex]\[
g = 28
\][/tex]
- Substitute [tex]\( g = 28 \)[/tex] back into [tex]\( r = \frac{5}{4}g \)[/tex]:
[tex]\[
r = \frac{5}{4} \times 28
\][/tex]
[tex]\[
r = 35
\][/tex]
Thus, the number of red counters originally in the bag is 35, and the number of green counters is 28.
