Let's solve the problem step-by-step.

Let the number of boys be \(B\) and the number of girls be \(G\).

According to the question, when 12 girls leave the group, the ratio of boys to girls becomes 2 : 1.

This can be expressed as:

\(\frac{B}{G - 12} = \frac{2}{1}\)

From this equation, we have:

\(B = 2(G - 12)\) ... (1)

Next, if 30 boys leave the group, the ratio of girls to boys becomes 3 : 1. This can be expressed as:

\(\frac{G - 12}{B - 30} = \frac{3}{1}\)

From this equation, we have:

\(G - 12 = 3(B - 30)\)

Expanding it, we get:

\(G - 12 = 3B - 90\)

Or, \(G = 3B - 78\) ... (2)

Now we have two equations:

• \(B = 2G - 24\) ... (from equation 1)
• \(G = 3B - 78\) ... (equation 2 from above)

Substituting the value of \(B\) from equation (2) into equation (1):

\(B = 2(3B - 78) - 24\)

This simplifies to:

\(B = 6B - 156 - 24\)

Simplification gives us:

\(5B = 180\)

\(B = 36\)

Substitute \(B = 36\) back into equation (2):

\(G = 3 \times 36 - 78\)

\(G = 108 - 78 = 30\)

Hence, the initial number of boys and girls are 36 and 30 respectively.

Correct Answer: 36 boys and 30 girls