A person on tour has Rs. 10800 for his expencses. If he extends his tour by 4 days, he has to cut down his daily expenses by Rs. 90. Find the original duration of the tour.

Explanation

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

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

### Step 2: Calculate Daily Expenses
The total amount available for expenses is Rs. 10800. Therefore, the daily expense can be expressed as:
\[ \text{Daily Expense} = \frac{10800}{x} \]

### Step 3: Set Up the New Daily Expense
If the person extends the tour by 4 days, the new duration becomes \( x + 4 \) days. The new daily expense will then be:
\[ \text{New Daily Expense} = \frac{10800}{x + 4} \]

### Step 4: Relate the New Daily Expense to the Original Daily Expense
According to the problem, if the tour is extended by 4 days, the daily expenses must be reduced by Rs. 90. Thus, we can set up the equation:
\[ \frac{10800}{x} - \frac{10800}{x + 4} = 90 \]

### Step 5: Solve the Equation
To solve this equation, we will first find a common denominator, which is \( x(x + 4) \):
\[ \frac{10800(x + 4) - 10800x}{x(x + 4)} = 90 \]

This simplifies to:
\[ \frac{10800 \cdot 4}{x(x + 4)} = 90 \]

Multiplying both sides by \( x(x + 4) \):
\[ 43200 = 90x(x + 4) \]

Dividing both sides by 90:
\[ 480 = x(x + 4) \]

### Step 6: Rearranging to Form a Quadratic Equation
Rearranging gives us:
\[ x^2 + 4x - 480 = 0 \]

### Step 7: Factor the Quadratic Equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to -480 and add to 4. The numbers are 24 and -20. Thus, we can factor the equation as:
\[ (x + 24)(x - 20) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives:
1. \( x + 24 = 0 \) → \( x = -24 \) (not valid since days cannot be negative)
2. \( x - 20 = 0 \) → \( x = 20 \)

### Conclusion
The original duration of the tour is \( \boxed{20} \) days.

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