In a group of 120 pupils, 80 can play guitar only and 25 can play piano only. How many pupils can play both instruments if there are 3 pupils who can play neither?
I bet you are learning about Venn diagrams
Draw 2 intersecting circles within a rectangle, and label them G and P
place x in the intersection of both circles
place 3 in the region inside the rectangle but outside of the circles.
Place 80 in the G circle not intersected by P
Place 25 in the P circle not intersected by G
solve:
x + 80 + 25 + 3 = 120 , and the mystery is solved.
Well, it seems we have quite the musical bunch! Let's see if we can unravel this musical mystery.
In total, we have 120 pupils, right? And we know that 80 can play the guitar only and 25 can play the piano only. Now, even though we have 3 pupils who can play neither, we can still figure out how many can play both instruments!
So, let me grab my imaginary conductor's baton and guide you through this calculation. If 80 can play the guitar only, and 25 can play the piano only, we need to subtract those numbers from the total number of pupils to find out how many can play both.
120 - (80 + 25) = 15
Voilà! We have 15 pupils who can play both the guitar and the piano, showing off their musical multitasking skills!
Looks like we've got a talented group of musicians in our hands, don't we? Keep those musical notes dancing, you bunch of rhythmic geniuses!
To find the number of pupils who can play both guitar and piano, we need to subtract the number of pupils who can play guitar only and the number of pupils who can play piano only from the total number of pupils in the group:
Total number of pupils = 120
Number of pupils who can play guitar only = 80
Number of pupils who can play piano only = 25
Let's find the number of pupils who can play both instruments.
Number of pupils who can play both guitar and piano = Total number of pupils - Number of pupils who can play guitar only - Number of pupils who can play piano only.
Number of pupils who can play both guitar and piano = 120 - 80 - 25
Number of pupils who can play both guitar and piano = 15
Therefore, there are 15 pupils who can play both guitar and piano.
To find out how many pupils can play both guitar and piano, we can use the principle of inclusion-exclusion.
Step 1: Calculate the number of pupils who can play either guitar or piano.
- We know that there are 120 pupils in total, and some pupils can play guitar only (80) and some can play piano only (25).
- To find the number of pupils who can play either guitar or piano, we add these two numbers (80 + 25 = 105).
- Therefore, 105 pupils can play either guitar or piano.
Step 2: Subtract the number of pupils who can play either guitar or piano from the total number of pupils to find the number of pupils who can play both instruments.
- Since there are 3 pupils who can play neither guitar nor piano, we subtract this number from the total number of pupils (120 - 3 = 117).
- However, the 117 pupils include those who can play either guitar or piano (105 of them).
- So, to find the number of pupils who can play both instruments, we subtract the number of pupils who can play either guitar or piano from the total number of pupils (117 - 105 = 12).
Therefore, there are 12 pupils who can play both guitar and piano.