Out of 10000 people surveyed, 3700 liked city A, 4000 liked city B and 5000 liked city C. 700 people liked A and B ,1200 liked A and C and 1000 liked B and C. Each person liked at least one city. Then find the number of persons liking at least two cities as a % of number of people liking exactly one city.

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the number of people who liked at least two cities and those who liked exactly one city.

### Step 1: Define the variables
Let:
- \( N(A) = 3700 \) (people who liked city A)
- \( N(B) = 4000 \) (people who liked city B)
- \( N(C) = 5000 \) (people who liked city C)
- \( N(A \cap B) = 700 \) (people who liked both A and B)
- \( N(A \cap C) = 1200 \) (people who liked both A and C)
- \( N(B \cap C) = 1000 \) (people who liked both B and C)
- \( N(A \cup B \cup C) = 10000 \) (total surveyed)

### Step 2: Use the inclusion-exclusion principle
We need to find \( N(A \cap B \cap C) \) (people who liked all three cities). According to the inclusion-exclusion principle:

\[ N(A \cup B \cup C) = N(A) + N(B) + N(C) - N(A \cap B) - N(A \cap C) - N(B \cap C) + N(A \cap B \cap C) \]

Substituting the known values:

\[ 10000 = 3700 + 4000 + 5000 - 700 - 1200 - 1000 + N(A \cap B \cap C) \]

### Step 3: Simplify the equation
Calculating the sum:

\[ 10000 = 3700 + 4000 + 5000 - 700 - 1200 - 1000 + N(A \cap B \cap C) \]
\[ 10000 = 12700 - 2900 + N(A \cap B \cap C) \]
\[ 10000 = 9800 + N(A \cap B \cap C) \]

### Step 4: Solve for \( N(A \cap B \cap C) \)
\[ N(A \cap B \cap C) = 10000 - 9800 = 200 \]

### Step 5: Calculate the number of people liking at least two cities
Now we can find the number of people who liked at least two cities:

\[ N(\text{at least 2}) = N(A \cap B) + N(A \cap C) + N(B \cap C) - 2N(A \cap B \cap C) \]
Substituting the values:

\[ N(\text{at least 2}) = 700 + 1200 + 1000 - 2 \times 200 \]
\[ N(\text{at least 2}) = 700 + 1200 + 1000 - 400 \]
\[ N(\text{at least 2}) = 2500 \]

### Step 6: Calculate the number of people liking exactly one city
Next, we calculate the number of people who liked exactly one city:

1. For city A:
\[ N(A \text{ only}) = N(A) - (N(A \cap B) + N(A \cap C) - N(A \cap B \cap C)) \]
\[ N(A \text{ only}) = 3700 - (700 + 1200 - 200) = 3700 - 1700 = 2000 \]

2. For city B:
\[ N(B \text{ only}) = N(B) - (N(A \cap B) + N(B \cap C) - N(A \cap B \cap C)) \]
\[ N(B \text{ only}) = 4000 - (700 + 1000 - 200) = 4000 - 1500 = 2500 \]

3. For city C:
\[ N(C \text{ only}) = N(C) - (N(A \cap C) + N(B \cap C) - N(A \cap B \cap C)) \]
\[ N(C \text{ only}) = 5000 - (1200 + 1000 - 200) = 5000 - 2000 = 3000 \]

Now, adding these gives us the total number of people who liked exactly one city:

\[ N(\text{exactly 1}) = N(A \text{ only}) + N(B \text{ only}) + N(C \text{ only}) = 2000 + 2500 + 3000 = 7500 \]

### Step 7: Calculate the percentage
Finally, we need to find the percentage of people liking at least two cities as a percentage of those liking exactly one city:

\[ \text{Percentage} = \left( \frac{N(\text{at least 2})}{N(\text{exactly 1})} \right) \times 100 \]
\[ \text{Percentage} = \left( \frac{2500}{7500} \right) \times 100 = \frac{1}{3} \times 100 = 33.33\% \]

### Final Answer
The number of persons liking at least two cities as a percentage of the number of people liking exactly one city is **33.33%**.