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 ratio of the number of people who went for both Movies and Shopping (but not Dine-out) to that who went for both Movies and Dine-out (but not Shopping)?

Options

A. `1:4`
B. `1:5`
C. `2:3`
D. `3:4`

Answer

To solve the problem step by step, we will analyze the given information and use it to find the required ratio.

### Step 1: Calculate the total number of people who dined out and went shopping.
- Total number of people = 2000
- Percentage who dined out = 60%
- Percentage who went shopping = 54%

Calculating the actual numbers:
- Number of people who dined out = 60% of 2000 = \( \frac{60}{100} \times 2000 = 1200 \)
- Number of people who went shopping = 54% of 2000 = \( \frac{54}{100} \times 2000 = 1080 \)

### Step 2: Identify the number of people who went for only shopping.
- It is given that 500 people went for only shopping.

### Step 3: Set up variables for the unknowns.
- Let \( x \) be the number of people who went for both Movies and Dine-out (but not Shopping).
- Let \( y \) be the number of people who went for both Shopping and Dine-out (but not Movies).
- The number of people who went for all three activities is equal to \( y \).

### Step 4: Use the given information to set up equations.
1. The number of people who went for only Movies = \( \frac{x}{2} \) (half of those who went for both Movies and Dine-out).
2. The number of people who went for both Shopping and Movies (but not Dine-out) = 100.

### Step 5: Calculate the total number of people in the Shopping circle.
- Total in Shopping circle = Only Shopping + Both Shopping and Dine-out + Both Shopping and Movies (not Dine-out) + All three activities
- This can be expressed as:
\[ 500 + y + 100 + y = 1080 \]
Simplifying gives:
\[ 500 + 2y + 100 = 1080 \implies 2y = 480 \implies y = 240 \]

### Step 6: Calculate the total number of people in the Dine-out circle.
- Total in Dine-out circle = Only Dine-out + Both Dine-out and Movies (not Shopping) + Both Dine-out and Shopping (not Movies) + All three activities
- This can be expressed as:
\[ 1200 = \frac{x}{2} + x + 240 + 240 \]
Simplifying gives:
\[ 1200 = \frac{x}{2} + x + 480 \implies 1200 - 480 = \frac{3x}{2} \implies 720 = \frac{3x}{2} \implies x = 480 \]

### Step 7: Find the required ratio.
- We need the ratio of the number of people who went for both Movies and Shopping (but not Dine-out) to those who went for both Movies and Dine-out (but not Shopping).
- Number of people who went for both Movies and Shopping (but not Dine-out) = 100.
- Number of people who went for both Movies and Dine-out (but not Shopping) = \( x = 480 \).

Thus, the ratio is:
\[ \text{Ratio} = \frac{100}{480} = \frac{5}{24} \]

### Final Answer:
The ratio of the number of people who went for both Movies and Shopping (but not Dine-out) to those who went for both Movies and Dine-out (but not Shopping) is \( \frac{5}{24} \).

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