Three Languages Students Count

Out of 100 students, 30 studied German, 28 studied German, 42 learned French, eight studied German and German, 10 German and German, and 5 German and German. How many students have learned all three languages?

Final Answer:

p =  23

Step-by-step explanation:


100 = f+n+s+5+10+8-2p
42 = f+5+10-p
30 = n+5+8-p
28=s+8+10-p

100 = f+n+s+5+10+8-2·p
42 = f+5+10-p
30 = n+5+8-p
28=s+8+10-p

f+n-2p+s = 77
f-p = 27
n-p = 17
p-s = -10

Row 2 - Row 1 → Row 2
f+n-2p+s = 77
-n+p-s = -50
n-p = 17
p-s = -10

Row 3 + Row 2 → Row 3
f+n-2p+s = 77
-n+p-s = -50
-s = -33
p-s = -10

Pivot: Row 3 ↔ Row 4
f+n-2p+s = 77
-n+p-s = -50
p-s = -10
-s = -33


s = -33/-1 = 33
p = -10+s = -10+33 = 23
n = -50-p+s/-1 = -50-23+33/-1 = 40
f = 77-n+2p-s = 77-40+2 · 23-33 = 50

f = 50
n = 40
p = 23
s = 33

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