Refer to the data below and answer the question that follow:
In the survey among students at all the IIMs, it was found that 48% preferred coffee, 54% liked tea and 64% smoked. Of the total, 28% liked coffee and tea, 32% smoked and drank tea and 30% smoked and drank coffee. Only 6% did none of these. If the total number of students is 2000 then find.
The ratio of the number of students who like only coffee to the number who like only tea is

Options

A. `5:3`
B. `8:9`
C. `2:3`
D. `3:2`

Answer

To find the ratio of the number of students who like only coffee to the number of students who like only tea, we will follow these steps:

### Step 1: Calculate the total percentages
We know the following from the problem:
- Students who like coffee = 48%
- Students who like tea = 54%
- Students who smoke = 64%
- Students who like both coffee and tea = 28%
- Students who smoke and drink tea = 32%
- Students who smoke and drink coffee = 30%
- Students who like none = 6%

First, we need to find the percentage of students who like all three (coffee, tea, and smoke).

### Step 2: Set up the equation
Let \( x \) be the percentage of students who like all three (coffee, tea, and smoke). The total percentage of students liking at least one of the three options can be represented as:

\[ (48 + 54 + 64 - 28 - 30 - 32 + x) + 6 = 100 \]

### Step 3: Solve for \( x \)
Substituting the values into the equation:

\[ (48 + 54 + 64 - 28 - 30 - 32 + x) + 6 = 100 \]
\[ (172 - 90 + x) + 6 = 100 \]
\[ (82 + x) + 6 = 100 \]
\[ x + 88 = 100 \]
\[ x = 12 \]

So, 12% of students like all three.

### Step 4: Calculate the exclusive percentages
Now we can find the percentages of students who like only coffee, only tea, and only smoke:

- **Only Coffee**:
\[ \text{Only Coffee} = 48 - (10 + 12 + 14) = 48 - 36 = 12\% \]

- **Only Tea**:
\[ \text{Only Tea} = 54 - (10 + 12 + 14) = 54 - 36 = 18\% \]

### Step 5: Find the ratio
Now we can find the ratio of students who like only coffee to those who like only tea:

\[ \text{Ratio} = \frac{\text{Only Coffee}}{\text{Only Tea}} = \frac{12}{18} = \frac{2}{3} \]

### Conclusion
Thus, the ratio of the number of students who like only coffee to the number who like only tea is **2:3**.

---