5 pens and 6 pencils together cost Rs 9 and 3 pens and 2 pencils cost Rs 5. Find the cost of 1 pen and 1 pencil.

Answer

To solve the problem step by step, we will set up equations based on the information given and then solve for the cost of one pen and one pencil.

### Step 1: Define Variables
Let:
- \( x \) = cost of one pen (in rupees)
- \( y \) = cost of one pencil (in rupees)

### Step 2: Set Up the Equations
From the problem statement, we have two conditions:
1. The cost of 5 pens and 6 pencils is Rs 9.
\[ 5x + 6y = 9 \quad \text{(Equation 1)} \]
2. The cost of 3 pens and 2 pencils is Rs 5.
\[ 3x + 2y = 5 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We will use the elimination method to solve these equations. First, we will multiply Equation 2 by 3 to align the coefficients of \( y \).

\[ 3(3x + 2y) = 3(5) \]
This gives us:
\[ 9x + 6y = 15 \quad \text{(Equation 3)} \]

### Step 4: Subtract Equation 1 from Equation 3
Now, we will subtract Equation 1 from Equation 3:
\[ (9x + 6y) - (5x + 6y) = 15 - 9 \]
This simplifies to:
\[ 4x = 6 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{6}{4} = \frac{3}{2} \]

### Step 6: Substitute \( x \) Back to Find \( y \)
Now that we have \( x \), we will substitute it back into one of the original equations to find \( y \). We will use Equation 2:
\[ 3\left(\frac{3}{2}\right) + 2y = 5 \]
This simplifies to:
\[ \frac{9}{2} + 2y = 5 \]
To eliminate the fraction, we can convert 5 into halves:
\[ \frac{9}{2} + 2y = \frac{10}{2} \]
Now, subtract \( \frac{9}{2} \) from both sides:
\[ 2y = \frac{10}{2} - \frac{9}{2} = \frac{1}{2} \]

### Step 7: Solve for \( y \)
Now, divide by 2:
\[ y = \frac{1}{2} \div 2 = \frac{1}{4} \]

### Final Results
- Cost of one pen \( x = \frac{3}{2} \) rupees
- Cost of one pencil \( y = \frac{1}{4} \) rupees

### Summary
The cost of one pen is Rs 1.50 and the cost of one pencil is Rs 0.25.