Mr. Ankit is on tour to siachin and he has Rs. 360 for his expenditure. If he exceeds his tour by 4 days. He must trim down his daily expenditure by Rs 3. For how many day is Mr. Ankit on tour ?

Options

A. 20
B. 22
C. 24
D. 26

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define Variables
Let \( X \) be the number of days Mr. Ankit is originally on tour, and \( Y \) be his daily expenditure.

### Step 2: Set Up the Equation
From the problem, we know that:
\[ X \times Y = 360 \]
This equation represents the total amount of money Mr. Ankit has for his tour.

### Step 3: Adjust for Extended Days
If Mr. Ankit extends his tour by 4 days, his new number of days becomes \( X + 4 \). To compensate for the extra days, he must reduce his daily expenditure by Rs. 3, so his new daily expenditure becomes \( Y - 3 \).

### Step 4: Set Up the New Equation
The new equation for the total expenditure after extending the tour is:
\[ (X + 4)(Y - 3) = 360 \]

### Step 5: Expand the New Equation
Expanding the equation gives:
\[ XY - 3X + 4Y - 12 = 360 \]

### Step 6: Substitute the First Equation
From the first equation, we know \( XY = 360 \). Substitute this into the expanded equation:
\[ 360 - 3X + 4Y - 12 = 360 \]
This simplifies to:
\[ -3X + 4Y - 12 = 0 \]
Rearranging gives:
\[ 4Y = 3X + 12 \quad \text{(1)} \]

### Step 7: Express \( Y \) in Terms of \( X \)
From the first equation \( XY = 360 \), we can express \( Y \) as:
\[ Y = \frac{360}{X} \quad \text{(2)} \]

### Step 8: Substitute \( Y \) in Equation (1)
Substituting equation (2) into equation (1):
\[ 4 \left(\frac{360}{X}\right) = 3X + 12 \]
This simplifies to:
\[ \frac{1440}{X} = 3X + 12 \]

### Step 9: Multiply through by \( X \)
To eliminate the fraction, multiply through by \( X \):
\[ 1440 = 3X^2 + 12X \]
Rearranging gives:
\[ 3X^2 + 12X - 1440 = 0 \]

### Step 10: Simplify the Quadratic Equation
Dividing the entire equation by 3:
\[ X^2 + 4X - 480 = 0 \]

### Step 11: Solve the Quadratic Equation
Using the quadratic formula \( X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 4, c = -480 \):
\[ X = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-480)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ X = \frac{-4 \pm \sqrt{16 + 1920}}{2} = \frac{-4 \pm \sqrt{1936}}{2} \]
Calculating \( \sqrt{1936} = 44 \):
\[ X = \frac{-4 \pm 44}{2} \]

### Step 12: Calculate Possible Values for \( X \)
Calculating the two possible values:
1. \( X = \frac{40}{2} = 20 \)
2. \( X = \frac{-48}{2} = -24 \) (not valid since days cannot be negative)

Thus, \( X = 20 \).

### Conclusion
Mr. Ankit is on tour for **20 days**.

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