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 at least two languages?

Options

A. 15
B. 30
C. 45
D. 55

Answer

To solve the problem step by step, we can use a systematic approach involving a Venn diagram to visualize the relationships between the students learning different languages.

### Step 1: Define Variables
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: Gather Given Information
- Students learning only Hindi = 23
- Students learning only English = 17
- Students learning only French = 11
- Students learning all three languages (Hindi, English, French) = 15
- Total students not learning any language = 28
- Total students learning French = 46

### Step 3: Set Up the Equation for French Learners
From the information given, we can set up the equation for the total number of students learning French:
\[ \text{Total French learners} = \text{Only French} + \text{Hindi and French} + \text{English and French} + \text{All three} \]
Substituting the known values:
\[ 46 = 11 + x + 3x + 15 \]
This simplifies to:
\[ 46 = 26 + 4x \]
\[ 4x = 46 - 26 \]
\[ 4x = 20 \]
\[ x = 5 \]

### Step 4: Calculate the Other Variables
Now that we have \( x \):
- Students learning Hindi and French but not English = \( x = 5 \)
- Students learning Hindi and English but not French = \( 2x = 10 \)
- Students learning English and French but not Hindi = \( 3x = 15 \)

### Step 5: Calculate Total Students Learning Each Language
Now we can summarize the total number of students learning at least two languages:
- Students learning only Hindi = 23
- Students learning only English = 17
- Students learning only French = 11
- Students learning Hindi and French but not English = 5
- Students learning Hindi and English but not French = 10
- Students learning English and French but not Hindi = 15
- Students learning all three languages = 15

### Step 6: Calculate Total Students Learning at Least Two Languages
To find the total number of students learning at least two languages, we add:
- Students learning Hindi and French but not English = 5
- Students learning Hindi and English but not French = 10
- Students learning English and French but not Hindi = 15
- Students learning all three languages = 15

Thus, the total is:
\[ 5 + 10 + 15 + 15 = 45 \]

### Final Answer
The total number of students learning at least two languages is **45**.

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