Study the following information carefully and answer the given questions.
Following data refers to total 2000 people who visited a mall on Saturday for the following activities: Movies, Dine- out and Shopping. A person may engage in one or more than one activity. `60%` of the total number of people dined out and `54%` of the total number of people went for Shopping (out of which 500 went for only Shopping).-The number of people who went for only Movies was half of that who went for both Movies and Dine-out -(but not shopping), and those who went for all activities are equal to those who went for both Shopping and Dine-out (but not Movies). 100 people went for both Shopping and Movies} (but not Dine-out). What is the number ofpeople who went for all the three activities?

Options

A. 200
B. 120
C. 160
D. 240

Answer

To solve the problem step by step, we will use the information provided and set up equations based on the activities of the people visiting the mall.

### Step 1: Determine the total number of people for each activity
- Total people = 2000
- People who dined out = 60% of 2000 = 0.60 * 2000 = 1200
- People who went for shopping = 54% of 2000 = 0.54 * 2000 = 1080

### Step 2: Define variables for different groups
Let:
- \( Y \) = Number of people who went for all three activities (Movies, Dine-out, Shopping)
- \( X \) = Number of people who went for both Movies and Dine-out but not Shopping
- \( A \) = Number of people who went for only Movies
- \( B \) = Number of people who went for both Shopping and Dine-out but not Movies
- \( C \) = Number of people who went for both Shopping and Movies but not Dine-out

### Step 3: Set up equations based on the given information
1. From the problem, we know that the number of people who went for only Movies was half of those who went for both Movies and Dine-out (but not Shopping):
\[ A = \frac{1}{2}X \]

2. The number of people who went for all activities is equal to those who went for both Shopping and Dine-out (but not Movies):
\[ Y = B \]

3. We know that 100 people went for both Shopping and Movies but not Dine-out:
\[ C = 100 \]

4. The total number of people who went for Shopping can be expressed as:
\[ 500 + Y + C + B = 1080 \]
Substituting \( C \) and \( B \):
\[ 500 + Y + 100 + Y = 1080 \]
Simplifying gives:
\[ 2Y + 600 = 1080 \implies 2Y = 480 \implies Y = 240 \]

### Step 4: Conclusion
The number of people who went for all three activities (Movies, Dine-out, Shopping) is \( Y = 240 \).