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Answers:

12 red and 9 green counters originally

Explanation:

r = number of red counters originally in the bag

g = number of green counters originally in the bag

We have g number of green counters out of g+r counters total. The probability of picking green is g/(g+r). We set it equal to 3/7 as this is the stated probability of picking green.

Let's solve for g

g/(g+r) = 3/7

7g = (g+r)*3 ... cross multiply

7g = 3g+3r

7g-3g = 3r

4g = 3r

g = 3r/4

g = (3/4)r

g = 0.75r

We'll use this equation later.

Whatever counter is selected is placed back into the bag, or it is replaced.

Now we add 2 more red and 3 more green counters.

The green goes from g to g+3, while the total goes from g+r to g+r+2+3 = g+r+5

The probability of picking green after these counters are added in is now (g+3)/(g+r+5) which is the same as 6/13.

We get the equation

(g+3)/(g+r+5) = 6/13

We'll plug in the previous equation we found (when we isolated g), and solve for r like so

(g+3)/(g+r+5) = 6/13

(0.75r+3)/(0.75r+r+5) = 6/13

(0.75r+3)/(1.75r+5) = 6/13

13(0.75r+3) = 6(1.75r+5) ... cross multiply

9.75r+39 = 10.5r+30

9.75r-10.5r = 30-39

-0.75r = -9

r = -9/(-0.75)

r = 12

There were originally 12 red counters in the bag

g = 0.75*r

g = 0.75*12

g = 9

And there were originally 9 green counters in the bag.

This would mean there were 9+12 = 21 counters total. The probability of picking green is 9/21 = (3*3)/(3*7) = 3/7 which matches with what the instructions are saying.

If we added 2 red and 3 green, then we have 9+3 = 12 green and 21+2+3 = 26 total. The probability of picking green now is 12/26 = (2*6)/(2*13) = 6/13. This also matches with what the instructions mention. Therefore, the answers have been confirmed.

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step-by-step explanation: