There are officer and Assistant in an office. If 15 Assistant went on leave then every Assistant has two officers, after that 45 officers went on leave now every officer has 5 Assistants. Then find out how many Assistant were there initially.

Options

A. 30
B. 40
C. 25
D. 35

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information provided.

### Step 1: Define Variables
Let:
- \( x \) = number of assistants initially
- \( y \) = number of officers initially

### Step 2: Set Up the First Condition
According to the problem, when 15 assistants go on leave, every assistant has 2 officers. This gives us the equation:
\[ y = \frac{x - 15}{2} \]
Rearranging this gives:
\[ x - 15 = 2y \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Condition
After 45 officers go on leave, every officer has 5 assistants. The number of officers remaining is \( y - 45 \) and the number of assistants remaining is \( x - 15 \). This gives us the equation:
\[ y - 45 = \frac{x - 15}{5} \]
Rearranging this gives:
\[ 5(y - 45) = x - 15 \]
Expanding and rearranging gives:
\[ 5y - 225 = x - 15 \]
Thus, we have:
\[ x - 5y = -210 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x - 2y = 30 \) (from Equation 1)
2. \( x - 5y = -210 \) (from Equation 2)

We can solve these equations simultaneously. Let's subtract Equation 1 from Equation 2:
\[ (x - 5y) - (x - 2y) = -210 - 30 \]
This simplifies to:
\[ -3y = -240 \]
Dividing both sides by -3 gives:
\[ y = 80 \]

### Step 5: Substitute Back to Find \( x \)
Now substitute \( y = 80 \) back into Equation 1:
\[ x - 2(80) = 30 \]
This simplifies to:
\[ x - 160 = 30 \]
Adding 160 to both sides gives:
\[ x = 190 \]

### Conclusion
The initial number of assistants is \( x = 190 \).