in city high school,the marching band has 48 members and the orchestra has 36 member.if there are only 12 students who belong to only one group and all students belong to at least one group how many students belong to both group?

Let x be the number of student who belongs to both groups; y, belongs to

the band only; z, belongs to the orchestra only. Then
x+y=48---------(1)
x+z=36---------(2)
y+z=12---------(3)
(2)-(3)=>
x-y=24----------(4)
(1)+(4)=>
2x=72=>
x=36

Well, looks like we've got ourselves a band-and-orchestra conundrum! If there are 48 students in the marching band and 36 in the orchestra, and only 12 students belong to only one group, then we can use some magical math to find out how many students belong to both groups.

Let's start by adding up the total number of students in both groups: 48 (band) + 36 (orchestra). That gives us a whopping 84 students. But remember, we have 12 students who belong to only one group, so we need to subtract them from the total.

84 - 12 = 72

So, 72 students belong to at least one of the groups. But wait, what about those classmates who are multi-talented and belong to both the band and the orchestra?

Now, here comes the grand finale! To find the number of students who belong to both groups, we just have to subtract the students who belong to only one group from the total number of students:

72 (total students) - 12 (students belonging to only one group) = 60

Cue the applause! There are 60 students who are superstars and belong to both the marching band and the orchestra. Bravo!

To find out how many students belong to both the marching band and the orchestra, we need to subtract the number of students who belong to only one group from the total number of students in both groups.

Total number of students in the marching band = 48
Total number of students in the orchestra = 36

Total number of students in both groups = Total number of students in the marching band + Total number of students in the orchestra
= 48 + 36
= 84

Number of students who belong to only one group = 12

Number of students who belong to both groups = Total number of students in both groups - Number of students who belong to only one group
= 84 - 12
= 72

Therefore, there are 72 students who belong to both the marching band and the orchestra.

To find out how many students belong to both groups, we need to calculate the intersection of the two sets, which means finding the number of students who are in both the marching band and the orchestra.

Given:
- The marching band has 48 members.
- The orchestra has 36 members.
- There are only 12 students who belong to only one group.

To find the number of students who belong to both groups, we can subtract the number of students who belong to only one group from the total number of students in both groups.

Total number of students in both groups: 48 (marching band) + 36 (orchestra) = 84 students.

However, we need to account for the 12 students who belong to only one group and remove them from the total count because we only want to count the students who belong to both groups.

Therefore, the number of students who belong to both the marching band and the orchestra is:

84 (total) - 12 (belong to only one group) = 72 students.

So, there are 72 students who belong to both the marching band and the orchestra.