Question

consider the following venn diagram, depicting the data from a survey of college students who were asked if they spoke another language than english, fluently.
data
spanish
168
36
3
14
french
65
39
german
81
12
a) how many students spoke both french and german fluently?
b) how many students spoke only french fluently?
c) how many students spoke spanish fluently?
d) how many students spoke both spanish and german, but not french?
e) how many students spoke both french and spanish, but not german?
f) how many students spoke only german?

Explanation:

Step1: Find students who spoke both French and German

In a Venn diagram, the intersection of French and German circles (excluding the part also in Spanish) gives the number. Here it is \(39 + 3=42\)

Step2: Find students who spoke only French

Only French is the part of French circle not overlapping with other circles. So \(65\)

Step3: Find students who spoke Spanish

Sum all parts in Spanish circle: \(168+36 + 3+14=221\)

Step4: Find students who spoke both Spanish and German, but not French

The intersection of Spanish and German circles not in French: \(14\)

Step5: Find students who spoke both French and Spanish, but not German

The intersection of French and Spanish circles not in German: \(36\)

Step6: Find students who spoke only German

Only German is the part of German circle not overlapping with other circles. So \(81\)

Answer:

a) \(42\)
b) \(65\)
c) \(221\)
d) \(14\)
e) \(36\)
f) \(81\)