In a group of 132 people 50, 60, 70 people like three different sweets-Barfi, Jalebi, Rasgulla, respectively. The number of people who like all the three sweets is half the number of people who like exactly two sweets. The number of people who like exactly any two out of the three sweets is the same as those who like exactly any other two of the three sweets. find the number of people who like Rasgulla or Jalebi but not Barfi

Options

A. 82
B. 42
C. 48
D. 38

Answer

To solve the problem step by step, we will use the information provided and set up equations based on the given conditions.

### Step 1: Define Variables
Let:
- \( B \) = Number of people who like Barfi = 50
- \( J \) = Number of people who like Jalebi = 60
- \( R \) = Number of people who like Rasgulla = 70
- \( G \) = Number of people who like all three sweets
- \( D \) = Number of people who like both Barfi and Jalebi (but not Rasgulla)
- \( F \) = Number of people who like both Barfi and Rasgulla (but not Jalebi)
- \( E \) = Number of people who like both Jalebi and Rasgulla (but not Barfi)

### Step 2: Set Up Equations
From the problem statement, we know:
1. The total number of people is 132.
2. The number of people who like all three sweets is half the number of people who like exactly two sweets:
\[ G = \frac{1}{2}(D + E + F) \]
3. The number of people who like exactly any two out of the three sweets is the same:
\[ D = E = F \]
Let \( D = E = F = x \).

### Step 3: Substitute and Simplify
Substituting \( D, E, F \) with \( x \):
- The equation for \( G \) becomes:
\[ G = \frac{1}{2}(x + x + x) = \frac{3x}{2} \]

### Step 4: Write Total Equations for Each Sweet
Using the total counts for each sweet:
1. For Barfi:
\[ B = D + F + G + \text{(only Barfi)} \implies 50 = x + x + \frac{3x}{2} + \text{(only Barfi)} \]
This simplifies to:
\[ 50 = 2x + \frac{3x}{2} + \text{(only Barfi)} \implies 50 = \frac{7x}{2} + \text{(only Barfi)} \]

2. For Jalebi:
\[ J = D + E + G + \text{(only Jalebi)} \implies 60 = x + x + \frac{3x}{2} + \text{(only Jalebi)} \]
This simplifies to:
\[ 60 = 2x + \frac{3x}{2} + \text{(only Jalebi)} \implies 60 = \frac{7x}{2} + \text{(only Jalebi)} \]

3. For Rasgulla:
\[ R = F + E + G + \text{(only Rasgulla)} \implies 70 = x + x + \frac{3x}{2} + \text{(only Rasgulla)} \]
This simplifies to:
\[ 70 = 2x + \frac{3x}{2} + \text{(only Rasgulla)} \implies 70 = \frac{7x}{2} + \text{(only Rasgulla)} \]

### Step 5: Solve for \( x \)
Now, we can express the total number of people in terms of \( x \):
\[ \text{(only Barfi)} + \text{(only Jalebi)} + \text{(only Rasgulla)} + D + E + F + G = 132 \]
Substituting the values:
\[ \text{(only Barfi)} + \text{(only Jalebi)} + \text{(only Rasgulla)} + 3x + \frac{3x}{2} = 132 \]

### Step 6: Solve the Equations
From the equations for Barfi, Jalebi, and Rasgulla, we can find the number of people who only like each sweet.

1. From Barfi:
\[ \text{(only Barfi)} = 50 - 2x - \frac{3x}{2} = 50 - \frac{7x}{2} \]
2. From Jalebi:
\[ \text{(only Jalebi)} = 60 - 2x - \frac{3x}{2} = 60 - \frac{7x}{2} \]
3. From Rasgulla:
\[ \text{(only Rasgulla)} = 70 - 2x - \frac{3x}{2} = 70 - \frac{7x}{2} \]

### Step 7: Substitute Back and Solve
Now we can substitute these back into the total equation and solve for \( x \):
\[ \left(50 - \frac{7x}{2}\right) + \left(60 - \frac{7x}{2}\right) + \left(70 - \frac{7x}{2}\right) + 3x + \frac{3x}{2} = 132 \]
Combine like terms and solve for \( x \).

### Step 8: Calculate the Values
After solving, we find:
- \( x = 8 \)
- \( G = 12 \)
- \( D = E = F = 8 \)
- Calculate the number of people who like Rasgulla or Jalebi but not Barfi:
\[ E + R = 8 + 42 = 50 \]

### Final Answer
The number of people who like Rasgulla or Jalebi but not Barfi is **82**.