3 pencils and 4 pens cost `Rs. 270` while 4 pencils and 3 pens cost Rs. 150 . Find the combined cost of one pencil and one pen.

Answer

To solve the problem step by step, we can set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( X \) = cost of one pencil (in Rs.)
- \( Y \) = cost of one pen (in Rs.)

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The cost of 3 pencils and 4 pens is Rs. 270:
\[ 3X + 4Y = 270 \quad \text{(Equation 1)} \]
2. The cost of 4 pencils and 3 pens is Rs. 150:
\[ 4X + 3Y = 150 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We can solve these two equations simultaneously. Let's use the method of elimination.

#### Step 3.1: Multiply the Equations
To eliminate one variable, we can multiply Equation 1 by 4 and Equation 2 by 3:
- Multiply Equation 1 by 4:
\[ 4(3X + 4Y) = 4(270) \implies 12X + 16Y = 1080 \quad \text{(Equation 3)} \]
- Multiply Equation 2 by 3:
\[ 3(4X + 3Y) = 3(150) \implies 12X + 9Y = 450 \quad \text{(Equation 4)} \]

#### Step 3.2: Subtract the Equations
Now, subtract Equation 4 from Equation 3:
\[ (12X + 16Y) - (12X + 9Y) = 1080 - 450 \]
This simplifies to:
\[ 7Y = 630 \]
Now, divide both sides by 7:
\[ Y = 90 \]

#### Step 3.3: Substitute Back to Find X
Now that we have \( Y \), we can substitute it back into one of the original equations to find \( X \). Let's use Equation 1:
\[ 3X + 4(90) = 270 \]
This simplifies to:
\[ 3X + 360 = 270 \]
Subtract 360 from both sides:
\[ 3X = 270 - 360 \]
\[ 3X = -90 \]
Now divide by 3:
\[ X = -30 \]

### Step 4: Find the Combined Cost
The combined cost of one pencil and one pen is:
\[ X + Y = -30 + 90 = 60 \]

### Final Answer
The combined cost of one pencil and one pen is Rs. 60.

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