let

g = the number of girls at the beginning of the year

b = the number of boys at the beginning of the year

the ratio of girls to boys is 5 to 7 in the junior high band:

g : b = 5 : 7 cross-multiply

7g = 5b

at the beginning of the year there were 72 students in the band:

g + b = 72

by solving the system of equations:

7g = 5b

g + b = 72

we find:

g = 30 girls

b = 42 boys

click here to see the step by step solution of the system of equations:

By the end of the year the ratio of girls to boys was 3 to 4 and there are 48 boys in the band.

let x represent the number of girls that joined the band than we have:

(30 + x) : 48 = 3 : 4 cross-multiply

4(30 + x) = 3*48

by solving we find:

x = 6 girls

click here to see the step by step solution of the equation:

6 girls joined the band during the school year.