In a school of 120 students, every student learns at least one of three languages French, German and Japanese. 70 students learn French, 55 learn German and 40 learn Japanese, and 8 students learn all three languages. How many students learn exactly two of the languages?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 29
When the three language totals are added, a student who learns one language is counted once, a student who learns exactly two languages twice, and a student who learns all three languages three times.
Let \(x\) students learn exactly one language and \(y\) students exactly two; 8 learn all three
Every student learns at least one: \(x+y+8=120\), so \(x+y=112\)
Sum of the totals: \(x+2y+3\times 8=70+55+40=165\), so \(x+2y=141\)
Subtracting: \(y=141-112=29\)
(Then \(x=112-29=83\) students learn exactly one language)
Checking the other options
Option (1): 83 — Incorrect. This is the number who learn exactly one language.
Option (2): 37 — Incorrect. This also counts the 8 who learn all three, so it is the number who learn at least two.
Option (3): 53 — Incorrect. This forgets that the 8 students who learn all three are counted three times in the totals: 165 - 112 = 53.