How many students are in the orchestra only?

Question

what is the answer to: at euclid middle school there are 33 in courus, 45 instudents in band and 21 in the orchesta. Fifteen students are in both the band and courus, 9 are in both orchastra and chourus 4 are in the band and orchastra 2 students are in all three.

Answer 1

Draw three circles that interlock each other, and label them C (chorus), B (band), O (orchestra). This is the Venn diagram. The interlocking circles divide space into 7 regions:

a. Chorus only

b. Band only

c. Orchestra only

d. C and B only

e. B and O only

f. C and O only

g. All 3
 

Now put in figures. Combined letters mean addition of their region numbers.

adfg=33; bdeg=45; cefg=21; dg=15; fg=9; eg=4; g=2.

From these we get:

e=2, because g=2 and ge=4; f=7; d=13; c=21-2-7-2=10; b=45-13-2-2=28; a=33-13-7-2=11.

So (a,b,c,d,e,f,g)=(11,28,10,13,2,7,2). There are c=10 on the orchestra only.