A certain number of capsules were purchased for ₹216, 15 more capsules could have been purchased in the same amount if each capsule was cheaper by ₹10. What was the number of capsules purchased?

Options

A. 6
B. 14
C. 8
D. 12

Explanation

To solve the problem step-by-step, we can follow this approach:

### Step 1: Define Variables
Let:
- \( Y \) = number of capsules purchased
- \( X \) = price of each capsule

### Step 2: Set Up the First Equation
According to the problem, the total cost of the capsules purchased is ₹216. Therefore, we can write the first equation as:
\[ Y \cdot X = 216 \quad \text{(1)} \]

### Step 3: Set Up the Second Equation
The problem states that if each capsule were cheaper by ₹10, then 15 more capsules could have been purchased for the same amount (₹216). The new price per capsule would be \( X - 10 \), and the new number of capsules would be \( Y + 15 \). Thus, we can write the second equation as:
\[ (Y + 15)(X - 10) = 216 \quad \text{(2)} \]

### Step 4: Expand the Second Equation
Expanding equation (2):
\[ Y \cdot X - 10Y + 15X - 150 = 216 \]
Substituting \( Y \cdot X = 216 \) from equation (1) into this equation:
\[ 216 - 10Y + 15X - 150 = 216 \]
Simplifying this gives:
\[ -10Y + 15X - 150 = 0 \]
Rearranging it, we get:
\[ 15X - 10Y = 150 \quad \text{(3)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( Y \cdot X = 216 \) (1)
2. \( 15X - 10Y = 150 \) (3)

From equation (1), we can express \( Y \) in terms of \( X \):
\[ Y = \frac{216}{X} \quad \text{(4)} \]

Substituting equation (4) into equation (3):
\[ 15X - 10\left(\frac{216}{X}\right) = 150 \]
Multiplying through by \( X \) to eliminate the fraction:
\[ 15X^2 - 2160 = 150X \]
Rearranging gives:
\[ 15X^2 - 150X - 2160 = 0 \]

### Step 6: Simplify the Quadratic Equation
Dividing the entire equation by 15:
\[ X^2 - 10X - 144 = 0 \]

### Step 7: Factor or Use the Quadratic Formula
We can factor this equation:
\[ (X - 18)(X + 8) = 0 \]
Thus, \( X = 18 \) or \( X = -8 \). Since price cannot be negative, we take:
\[ X = 18 \]

### Step 8: Find the Number of Capsules
Substituting \( X \) back into equation (4) to find \( Y \):
\[ Y = \frac{216}{18} = 12 \]

### Conclusion
The number of capsules purchased is \( \boxed{12} \).

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