Ther are two biludings P and Q. If 15 persons are sent from P to Q, then the number of per sons in each biluding is the same. If 20 persons are sent from Q to P, then the number of persons in P is double the number of persons in Q. How many persons are there in building P?

Options

A. 80
B. 140
C. 120
D. 90

Answer

Let's solve the problem step by step.

### Step 1: Define Variables
Let the number of persons in building P be \( P \) and the number of persons in building Q be \( Q \).

**Hint:** Start by defining the variables for the unknowns in the problem.

### Step 2: Set Up the First Equation
According to the problem, if 15 persons are sent from P to Q, then the number of persons in each building becomes equal. This can be expressed as:
\[ P - 15 = Q + 15 \]
Rearranging this gives:
\[ P = Q + 30 \quad \text{(Equation 1)} \]

**Hint:** Use the information about the equal number of persons after transferring to form your first equation.

### Step 3: Set Up the Second Equation
Next, if 20 persons are sent from Q to P, then the number of persons in P becomes double the number of persons in Q. This can be expressed as:
\[ P + 20 = 2(Q - 20) \]
Expanding this gives:
\[ P + 20 = 2Q - 40 \]
Rearranging this gives:
\[ P - 2Q = -60 \quad \text{(Equation 2)} \]

**Hint:** Use the second scenario to create another equation that relates P and Q.

### Step 4: Substitute Equation 1 into Equation 2
Now, substitute \( P \) from Equation 1 into Equation 2:
\[ (Q + 30) - 2Q = -60 \]
This simplifies to:
\[ 30 - Q = -60 \]
Rearranging gives:
\[ Q = 90 \]

**Hint:** Substitute the expression for one variable into the other equation to find the value of one variable.

### Step 5: Find the Value of P
Now that we have \( Q \), substitute it back into Equation 1 to find \( P \):
\[ P = Q + 30 = 90 + 30 = 120 \]

**Hint:** Use the value of the variable you just found to calculate the other variable.

### Conclusion
The number of persons in building P is \( 120 \).

**Final Answer:** There are 120 persons in building P.