You have a bag which contains only red and green marbles. In this bag with $x^{2} 5$ marbles total, $x 1$ are red. Also, $x-3$ marbles have a scratch on them. The probability of drawing a red marble from the original bag is equal to that of drawing a marble with a scratch from the marbles left in the bag after twenty scratch-free marbles are taken out of the full bag. How many marbles were originally in the bag?

The original number of marbles in the bag is 30.

The given parameters are:

The probability of red is:

When 20 marbles were removed, the marbles left are:

This gives:

The probability of selecting a scratched marble at this point is:

This probability equals the probability of red.

i.e.

So, we have:

Cross multiply

Expand

Subtract from both sides

Add 15 to both sides

Collect like terms

Divide through by 4

Expand

Split

or

or

x can't be 0.

So:

The number of marbles initially is:

Hence, the original number of marbles in the bag is 30.

Read more about probabilities at: