The Students of a class are offered three language (Hindi. English and French). 15 students learn all the three language whereas 28 students do not learn any language. The number of students learning Hindi and English but not French is twice the number of students learning Hindi and French but not English. The number of students learning English and French but not Hindi is thrice the number of students learning Hindi and French but not English. 23 students learn only Hindi and 17 students learn only English. The total number of students learning French is 46 and the total number of students learning only French is 11.
How many students learn precisely two languages?

Options

A. A. 55
B. B. 40
C. C. 30
D. D. 13

Answer

To solve the problem, we will use a Venn diagram to represent the relationships between the students learning Hindi, English, and French. Let's break down the information step by step.

### Step 1: Define Variables
1. Let:
- \( x \) = number of students learning Hindi and French but not English.
- \( 2x \) = number of students learning Hindi and English but not French.
- \( 3x \) = number of students learning English and French but not Hindi.

### Step 2: Fill in Known Values
2. From the problem, we know:
- 15 students learn all three languages (Hindi, English, and French).
- 23 students learn only Hindi.
- 17 students learn only English.
- 11 students learn only French.
- 28 students do not learn any language.
- The total number of students learning French is 46.

### Step 3: Set Up the Equation for French Learners
3. The total number of students learning French can be expressed as:
\[ \text{Total French} = \text{Only French} + \text{Hindi and French but not English} + \text{English and French but not Hindi} + \text{All three} \]
Substituting the known values:
\[ 46 = 11 + x + 3x + 15 \]
Simplifying this gives:
\[ 46 = 11 + 4x + 15 \]
\[ 46 = 26 + 4x \]
\[ 20 = 4x \]
\[ x = 5 \]

### Step 4: Calculate Other Values
4. Now we can calculate the number of students in each category:
- Students learning Hindi and English but not French: \( 2x = 2 \times 5 = 10 \)
- Students learning English and French but not Hindi: \( 3x = 3 \times 5 = 15 \)

### Step 5: Summarize All Values
5. We can summarize the number of students in each category:
- Only Hindi: 23
- Only English: 17
- Only French: 11
- Hindi and English but not French: 10
- Hindi and French but not English: 5
- English and French but not Hindi: 15
- All three languages: 15
- No language: 28

### Step 6: Calculate Students Learning Precisely Two Languages
6. The total number of students learning precisely two languages is:
\[ \text{Precisely two languages} = (Hindi \cap English \cap \neg French) + (Hindi \cap French \cap \neg English) + (English \cap French \cap \neg Hindi) \]
Substituting the values:
\[ = 10 + 5 + 15 = 30 \]

### Final Answer
Thus, the number of students who learn precisely two languages is **30**.

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