This solution is only if you are comfortable with using algebra method.
First, we must understand that the number of people in both groups are equal. As seen in the model, the number of people in 1st group = X(boys) + 20 + X(girls) = 2X + 20. The number of people in the 2nd group = Y(boys)+Y(girls)+12 = 2Y+12... So 2X+20 = 2Y+12. Using algebra manipulation, we reduce the equation to 4=Y-X.

Equation 1 : 4=Y-X

Second, 45% of all the participants are girls. The number of participants = 1st grp + 2nd grp = (2X +20) + (2Y+12) = 2X + 2Y + 32. 45% = 9/20. Therefore, (9/20) * (2X + 2Y +32) = number of girls = X+Y+12. Further manipulation, we reached our next equation, X=24-Y.

Equation 2: X=24-Y

Next, we substitute our second equation into the 1st equation, and we get 4=Y-X = Y- (24-Y) = 2Y-24. Solving this, we get 2Y = 28 and Y=14. Since X=24-Y, X = 24-14=10.

X=10
Y=14

With X and Y, we can now find the number of boys = (X+20) (-from 1st group + Y (-from 2nd group) = X+20+Y= 10+20+14 =44.

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difference unchanged