There are two examination halls P and Q. If 10 students shifted P to Q, then the number of students will be equal in both the examination halls. If 20 students shifted from Q to P,then the students of P would be doubled to the students of Q. The numbers of students would be in P and Q, respectively are

Options

A. 60, 40
B. 70, 50
C. 80, 60
D. 100, 80

Answer

To solve the problem, we will define the number of students in examination halls P and Q as follows:

Let:
- \( X \) = number of students in hall P
- \( Y \) = number of students in hall Q

We will set up equations based on the conditions provided in the problem.

### Step 1: Set up the first equation
According to the first condition, if 10 students are shifted from P to Q, the number of students in both halls will be equal. This can be expressed as:
\[ X - 10 = Y + 10 \]
Rearranging this gives:
\[ X - Y = 20 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation
According to the second condition, if 20 students are shifted from Q to P, the number of students in P will be double that in Q. This can be expressed as:
\[ X + 20 = 2(Y - 20) \]
Expanding this gives:
\[ X + 20 = 2Y - 40 \]
Rearranging this gives:
\[ X - 2Y = -60 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( X - Y = 20 \) (Equation 1)
2. \( X - 2Y = -60 \) (Equation 2)

We can solve these equations simultaneously. We can start by expressing \( X \) from Equation 1:
\[ X = Y + 20 \]

Now, substitute \( X \) in Equation 2:
\[ (Y + 20) - 2Y = -60 \]
Simplifying this gives:
\[ 20 - Y = -60 \]
Rearranging gives:
\[ -Y = -60 - 20 \]
\[ -Y = -80 \]
Thus, we find:
\[ Y = 80 \]

### Step 4: Find the value of X
Now that we have \( Y \), we can substitute it back into Equation 1 to find \( X \):
\[ X - 80 = 20 \]
\[ X = 20 + 80 \]
\[ X = 100 \]

### Final Answer
The number of students in examination halls P and Q are:
- Hall P (X) = 100
- Hall Q (Y) = 80