Question

In a school where there was a compulsion to learn at least one foreign language from the choices given, namely German, French and Spanish: 28 students took French, 30 took German, and 32 took Spanish. 6 students learnt French and German, 8 students learnt German and Spanish, and 10 students learnt French and Spanish. 54 students learnt only one foreign language, while 20 students learnt only German. Find the number of students in the school.

• A. 60
• B. 62
• C. 70
• D. None of the above

Hint:

Split into only-one, exactly-two, and all-three regions, and use the given sums to solve for each unknown.

Answer 1

The Correct Option is C

To find the total number of students in the school, we need to consider all students learning French, German, and Spanish, and account for those learning multiple languages. This is a typical problem involving set theory and can be solved using the principle of inclusion-exclusion.

Let: 

• \(F\) be the set of students learning French.
• \(G\) be the set of students learning German.
• \(S\) be the set of students learning Spanish.

We are given:

• \(|F| = 28\) (students took French)
• \(|G| = 30\) (students took German)
• \(|S| = 32\) (students took Spanish)
• \(|F \cap G| = 6\) (students learnt both French and German)
• \(|G \cap S| = 8\) (students learnt both German and Spanish)
• \(|F \cap S| = 10\) (students learnt both French and Spanish)
• \(|F \cup G \cup S| = x\) (total number of students learning at least one language)
• \(|F \cap G \cap S| = y\) (students learning all three languages, value unknown)
• 54 students learnt only one foreign language.
• 20 students learnt only German.

First, note that students learning only German is already known: \(|G| - |F \cap G| - |G \cap S| + |F \cap G \cap S| = 20\).

Using the principle of inclusion-exclusion for three sets, the formula for the total number of students who learned at least one language is given by:

\(|F \cup G \cup S| = |F| + |G| + |S| - |F \cap G| - |G \cap S| - |F \cap S| + |F \cap G \cap S|\)

Substituting the known values:

\(x = 28 + 30 + 32 - 6 - 8 - 10 + y\)

Given 54 students learnt only one language:

\(54 = |F| + |G| + |S| - 2(|F \cap G| + |G \cap S| + |F \cap S|) + 3|F \cap G \cap S|\)

Rewriting and substituting gives us:

\(54 = 28 + 20 + (32 - 10 - 8) = 62 - y\)

This yields \(y = 2\) (students learning all three languages).

Substitute \(y = 2\) back in the total students equation:

\(x = 28 + 30 + 32 - 6 - 8 - 10 + 2 = 70\)

Therefore, the total number of students in the school is: 70.

Answer 2

Step 1: Set up the Venn diagram regions.
Let a, b, c be the "only one language" regions for German, French, Spanish. Let d, e, f be the "exactly two languages" regions: d for German and French only, e for German and Spanish only, f for French and Spanish only. Let g be the region for all three languages. We are told a = 20 (only German).

Step 2: Use the "only one language" total.
Exactly one language covers 54 students, so a + b + c = 54. Since a = 20, this gives b + c = 34.

Step 3: Use the total counts for each language.
Each language's total counts its "only" region plus both "exactly two" regions touching it plus the "all three" region. Adding all three language totals: German + French + Spanish = (a+b+c) + 2(d+e+f) + 3g = 30+28+32 = 90. Let X = a+b+c = 54, Y = d+e+f, Z = g. So X + 2Y + 3Z = 90, which gives 2Y + 3Z = 90 - 54 = 36.

Step 4: Use the pairwise overlap data.
"French and German" of 6 counts d + g = 6 (people in both, whether or not they also take Spanish). Similarly "German and Spanish" gives e + g = 8, and "French and Spanish" gives f + g = 10. Adding these three: (d+e+f) + 3g = 24, which is Y + 3Z = 24.

Step 5: Solve the two equations for Y and Z.
We have 2Y + 3Z = 36 and Y + 3Z = 24. Subtracting the second from the first gives Y = 12. Then 3Z = 24 - 12 = 12, so Z = 4.

Step 6: Add up the total.
Since every student learns at least one language, nobody falls in a "none" region. Total students = X + Y + Z = 54 + 12 + 4 = 70.

Final Answer:
The number of students in the school is 70. \[ \boxed{70} \]