5% of the passengers do not like coffee, tea and lassi and 10% like all the three, 20% like coffee and tea, 25% like lassi and coffee and 25% like lassi and tea. 55% like coffee, 50% like tea, and 50% like lassi.
If the number of passengers is 180, then the number of passengers who like lassi only, is

Options

A. `10`
B. `18`
C. `27`
D. `36`

Answer

To solve the problem step by step, we will use the information provided and apply set theory concepts, particularly Venn diagrams.

### Step 1: Understand the total number of passengers
We are given that the total number of passengers is 180.

### Step 2: Calculate the percentages
We know:
- 5% of passengers do not like coffee, tea, or lassi.
- 10% like all three (coffee, tea, and lassi).
- 20% like coffee and tea.
- 25% like lassi and coffee.
- 25% like lassi and tea.
- 55% like coffee.
- 50% like tea.
- 50% like lassi.

### Step 3: Calculate the number of passengers for each percentage
1. **Passengers who do not like any of the three drinks**:
\[ 5\% \text{ of } 180 = 0.05 \times 180 = 9 \text{ passengers} \]

2. **Passengers who like all three drinks**:
\[ 10\% \text{ of } 180 = 0.10 \times 180 = 18 \text{ passengers} \]

3. **Passengers who like coffee and tea**:
\[ 20\% \text{ of } 180 = 0.20 \times 180 = 36 \text{ passengers} \]

4. **Passengers who like lassi and coffee**:
\[ 25\% \text{ of } 180 = 0.25 \times 180 = 45 \text{ passengers} \]

5. **Passengers who like lassi and tea**:
\[ 25\% \text{ of } 180 = 0.25 \times 180 = 45 \text{ passengers} \]

6. **Passengers who like coffee**:
\[ 55\% \text{ of } 180 = 0.55 \times 180 = 99 \text{ passengers} \]

7. **Passengers who like tea**:
\[ 50\% \text{ of } 180 = 0.50 \times 180 = 90 \text{ passengers} \]

8. **Passengers who like lassi**:
\[ 50\% \text{ of } 180 = 0.50 \times 180 = 90 \text{ passengers} \]

### Step 4: Set up the Venn diagram
Let:
- \( x \) = number of passengers who like only coffee
- \( y \) = number of passengers who like only tea
- \( z \) = number of passengers who like only lassi
- \( a \) = number of passengers who like coffee and tea but not lassi
- \( b \) = number of passengers who like coffee and lassi but not tea
- \( c \) = number of passengers who like tea and lassi but not coffee

From the information:
- \( a + 18 + b = 36 \) (for coffee and tea)
- \( b + 18 + c = 45 \) (for lassi and coffee)
- \( c + 18 + a = 45 \) (for lassi and tea)

### Step 5: Solve the equations
1. From \( a + 18 + b = 36 \):
\[ a + b = 36 - 18 = 18 \quad (1) \]

2. From \( b + 18 + c = 45 \):
\[ b + c = 45 - 18 = 27 \quad (2) \]

3. From \( c + 18 + a = 45 \):
\[ c + a = 45 - 18 = 27 \quad (3) \]

Now we have three equations:
- \( a + b = 18 \) (1)
- \( b + c = 27 \) (2)
- \( c + a = 27 \) (3)

### Step 6: Solve for \( a, b, c \)
From equation (1):
\[ b = 18 - a \]

Substituting \( b \) in equation (2):
\[ (18 - a) + c = 27 \implies c = 27 - 18 + a = 9 + a \quad (4) \]

Substituting \( c \) in equation (3):
\[ (9 + a) + a = 27 \implies 2a + 9 = 27 \implies 2a = 18 \implies a = 9 \]

Now substituting \( a \) back to find \( b \) and \( c \):
\[ b = 18 - 9 = 9 \]
\[ c = 9 + 9 = 18 \]

### Step 7: Find the number of passengers who like only lassi
Now we can find the number of passengers who like only lassi:
\[ \text{Total passengers who like lassi} = z + b + c + 18 = 90 \]
Substituting the values:
\[ z + 9 + 18 + 18 = 90 \implies z + 45 = 90 \implies z = 45 \]

### Step 8: Calculate the number of passengers who like only lassi in the total population
To find the number of passengers who like only lassi out of 180:
\[ \text{Passengers who like only lassi} = \frac{45}{100} \times 180 = 81 \]

### Final Answer
Thus, the number of passengers who like only lassi is **81**.

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