Pintu takes 6 days more than those of Nishu to complete certain work. If they work togeter they finish it in 4 days. How many days would it take to complete the work if they work alone.

Answer

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

### Step 1: Define Variables
Let \( x \) be the number of days Nishu takes to complete the work. According to the problem, Pintu takes 6 days more than Nishu, so he takes \( x + 6 \) days.

### Step 2: Set Up the Work Equation
When they work together, they complete the work in 4 days. Therefore, the work done by both Nishu and Pintu in one day can be expressed as:
\[ \frac{1}{x} + \frac{1}{x + 6} = \frac{1}{4} \]

### Step 3: Clear the Denominators
To eliminate the fractions, we can multiply through by \( 4x(x + 6) \):
\[ 4(x + 6) + 4x = x(x + 6) \]

### Step 4: Expand and Rearrange
Expanding both sides gives:
\[ 4x + 24 + 4x = x^2 + 6x \]
Combining like terms results in:
\[ 8x + 24 = x^2 + 6x \]
Rearranging the equation leads to:
\[ x^2 - 2x - 24 = 0 \]

### Step 5: Factor the Quadratic Equation
Now we need to factor the quadratic equation \( x^2 - 2x - 24 = 0 \). We look for two numbers that multiply to \(-24\) and add to \(-2\). The numbers are \( -6 \) and \( 4 \):
\[ (x - 6)(x + 4) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 6 = 0 \) → \( x = 6 \)
2. \( x + 4 = 0 \) → \( x = -4 \) (not possible since days cannot be negative)

Thus, \( x = 6 \).

### Step 7: Calculate Days for Pintu
Since Nishu takes \( x = 6 \) days, Pintu takes:
\[ x + 6 = 6 + 6 = 12 \text{ days} \]

### Conclusion
Nishu would take 6 days to complete the work alone, and Pintu would take 12 days to complete the work alone.

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