A person on tour has 360 for his expenses. He decides to extend his tour programme by 4 days which leads to cutting down daily expenses by 3 a day. The number of days of his tour programme is

Options

A. 15
B. 20
C. 18
D. 16

Answer

To solve the problem step by step, we will define variables and set up an equation based on the information provided.

### Step 1: Define Variables
Let \( x \) be the original number of days of the tour.

### Step 2: Set Up the Equation
The total amount of money available for expenses is 360 rupees. Therefore, the daily expenses can be expressed as:
\[ \text{Daily expenses} = \frac{360}{x} \]

If the person extends the tour by 4 days, the new number of days becomes \( x + 4 \). The new daily expenses after extending the tour and cutting down daily expenses by 3 rupees will be:
\[ \text{New daily expenses} = \frac{360}{x + 4} \]

### Step 3: Create the Equation
According to the problem, the new daily expenses are 3 rupees less than the original daily expenses. Therefore, we can set up the equation:
\[ \frac{360}{x + 4} = \frac{360}{x} - 3 \]

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply both sides of the equation by \( x(x + 4) \):
\[ 360x = 360(x + 4) - 3x(x + 4) \]

### Step 5: Expand and Simplify
Expanding both sides:
\[ 360x = 360x + 1440 - 3x^2 - 12x \]
Now, simplify the equation:
\[ 0 = 1440 - 3x^2 - 12x \]
Rearranging gives:
\[ 3x^2 + 12x - 1440 = 0 \]

### Step 6: Divide the Equation
To simplify, divide the entire equation by 3:
\[ x^2 + 4x - 480 = 0 \]

### Step 7: Factor the Quadratic Equation
Now, we will factor the quadratic equation:
\[ (x + 24)(x - 20) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
\[ x + 24 = 0 \quad \text{or} \quad x - 20 = 0 \]
This results in:
\[ x = -24 \quad \text{(not valid since days cannot be negative)} \quad \text{or} \quad x = 20 \]

### Conclusion
The original number of days of the tour program is \( x = 20 \).