Study the following information given in the paragraph carefully and answer the questions given below:
There are 1000 student in a college. Out of 1000 student some appeared in exams X,Y and Z while some did not. The number of student not appeared in any exam is equal to the number of student appeared in exam Z only. The number of students appeared in exam Y is 360. Ratio of the number of student appeared in exams X and Y only to number of students appeared in exams Y and Z only is 2:3. The number of students appeared in exams X and Z both is half of the number of student appeared in only exam Z. The number of students appeared in exam X only is `50%` more then the number of students appeared in Y only. The number of students appeared in all the three exams is `4%` of the total number of student in the college. The number of student appeared in Y exam only is equal to the no of students appeared in exam Y and Z only.
How many students have appeared in exam X or in exam Z?

Options

A. 240
B. 360
C. 500
D. 680

Answer

To solve the problem step by step, we will define variables and use the information provided in the question to derive the required values.

### Step 1: Define Variables
Let:
- \( a \) = number of students who appeared in exam X only
- \( b \) = number of students who appeared in exam Y only
- \( c \) = number of students who appeared in exam Z only
- \( d \) = number of students who appeared in exams X and Y only
- \( e \) = number of students who appeared in exams Y and Z only
- \( f \) = number of students who appeared in exams X and Z only
- \( g \) = number of students who appeared in all three exams X, Y, and Z

### Step 2: Analyze Given Information
1. The total number of students is 1000.
2. The number of students who did not appear in any exam is equal to the number of students who appeared in exam Z only: \( c \).
3. The number of students who appeared in exam Y is 360: \( b + e + d + g = 360 \).
4. The ratio of students who appeared in exams X and Y only to those who appeared in exams Y and Z only is \( \frac{d}{e} = \frac{2}{3} \).
5. The number of students who appeared in exams X and Z both is half of those who appeared in only exam Z: \( f = \frac{1}{2}c \).
6. The number of students who appeared in exam X only is 50% more than those who appeared in exam Y only: \( a = 1.5b \).
7. The number of students who appeared in all three exams is 4% of the total number of students: \( g = 0.04 \times 1000 = 40 \).
8. The number of students who appeared in exam Y only is equal to the number of students who appeared in exams Y and Z only: \( b = e \).

### Step 3: Set Up Equations
From the information above, we can set up the following equations:
1. \( b + e + d + g = 360 \) (from point 3)
2. \( d = \frac{2}{3}e \) (from point 4)
3. \( f = \frac{1}{2}c \) (from point 5)
4. \( a = 1.5b \) (from point 6)
5. \( g = 40 \) (from point 7)
6. \( b = e \) (from point 8)

### Step 4: Substitute and Solve
Substituting \( g = 40 \) and \( b = e \) into equation 1:
\[ b + b + d + 40 = 360 \]
\[ 2b + d + 40 = 360 \]
\[ 2b + d = 320 \]
Now substitute \( d = \frac{2}{3}b \):
\[ 2b + \frac{2}{3}b = 320 \]
To eliminate the fraction, multiply the entire equation by 3:
\[ 6b + 2b = 960 \]
\[ 8b = 960 \]
\[ b = 120 \]

Now substitute \( b \) back to find \( e \):
\[ e = b = 120 \]

Now find \( d \):
\[ d = \frac{2}{3}e = \frac{2}{3} \times 120 = 80 \]

### Step 5: Find \( a \), \( f \), and \( c \)
Now we can find \( a \):
\[ a = 1.5b = 1.5 \times 120 = 180 \]

Next, we can find \( c \) using the total number of students:
The total number of students who appeared in any exam is:
\[ a + b + c + d + e + f + g = 1000 - c \]
Substituting known values:
\[ 180 + 120 + c + 80 + 120 + f + 40 = 1000 - c \]
Using \( f = \frac{1}{2}c \):
\[ 180 + 120 + c + 80 + 120 + \frac{1}{2}c + 40 = 1000 - c \]
Combine like terms:
\[ 540 + \frac{3}{2}c = 1000 - c \]
Rearranging gives:
\[ \frac{3}{2}c + c = 1000 - 540 \]
\[ \frac{5}{2}c = 460 \]
\[ c = \frac{460 \times 2}{5} = 184 \]

Now find \( f \):
\[ f = \frac{1}{2}c = \frac{1}{2} \times 184 = 92 \]

### Step 6: Calculate Students Appeared in Exam X or Z
To find the total number of students who appeared in exam X or exam Z:
\[ \text{Total in X or Z} = a + f + c + g \]
Substituting the values:
\[ = 180 + 92 + 184 + 40 = 496 \]

### Final Answer
The number of students who have appeared in exam X or in exam Z is **496**.