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 English and French?

Options

A. 30
B. 43
C. 45
D. 73

Answer

To solve the problem step by step, we will use a systematic approach to analyze the information given and derive the required values.

### Step 1: Define Variables
Let's define the variables based on the information provided:
- Let \( x \) be the number of students learning Hindi and French but not English.
- The number of students learning Hindi and English but not French will then be \( 2x \) (as given).
- The number of students learning English and French but not Hindi will be \( 3x \) (as given).

### Step 2: List Known Values
From the problem, we know:
- Students learning only Hindi = 23
- Students learning only English = 17
- Students learning only French = 11
- Students learning all three languages = 15
- Students not learning any language = 28
- Total students learning French = 46

### Step 3: Set Up the Equation for French Learners
The total number of students learning French can be expressed as:
\[ \text{Only French} + \text{Hindi and French (not English)} + \text{English and French (not Hindi)} + \text{All three languages} = 46 \]
Substituting the known values:
\[ 11 + x + 3x + 15 = 46 \]
This simplifies to:
\[ 11 + 15 + 4x = 46 \]
\[ 26 + 4x = 46 \]

### Step 4: Solve for \( x \)
Now, isolate \( x \):
\[ 4x = 46 - 26 \]
\[ 4x = 20 \]
\[ x = 5 \]

### Step 5: Calculate Other Values
Now that we have \( x = 5 \):
- 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 6: Verify Total Students Learning French
Now, let's verify the total number of students learning French:
\[ \text{Only French} + \text{Hindi and French (not English)} + \text{English and French (not Hindi)} + \text{All three languages} = 46 \]
Substituting the values:
\[ 11 + 5 + 15 + 15 = 46 \]
This confirms our calculations are correct.

### Step 7: Find Total Students Learning English and French
To find the total number of students learning English and French, we add:
- Students learning only English = 17
- Students learning English and French but not Hindi = 15
- Students learning all three languages = 15

Thus, the total is:
\[ 17 + 15 + 15 = 47 \]

### Conclusion
Therefore, the total number of students learning English and French is **47**.