`(x-2)` men can do a piece of work in x days and `(x+7)` men can do `75%` of the same work in `(x-10)` days. Then in how many days can `(x+10)` men finish the work?

Options

A. 27 days
B. 12 days
C. 25 days
D. 18 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Work Done
Let the total work be denoted as \( W \).

From the problem, we know:
- \( (x-2) \) men can complete the work in \( x \) days.
- Therefore, the work done by \( (x-2) \) men in \( x \) days is:
\[ W = (x-2) \times x \]

### Step 2: Work Done by \( (x+7) \) Men
Next, we know that \( (x+7) \) men can complete \( 75\% \) of the work in \( (x-10) \) days. Thus:
\[ 0.75W = (x+7) \times (x-10) \]

### Step 3: Substitute \( W \) in the Second Equation
Substituting \( W \) from Step 1 into the second equation:
\[ 0.75 \times (x-2) \times x = (x+7) \times (x-10) \]

### Step 4: Expand Both Sides
Expanding both sides:
- Left side:
\[ 0.75(x^2 - 2x) = 0.75x^2 - 1.5x \]
- Right side:
\[ (x+7)(x-10) = x^2 - 10x + 7x - 70 = x^2 - 3x - 70 \]

### Step 5: Set the Equation
Setting the two sides equal:
\[ 0.75x^2 - 1.5x = x^2 - 3x - 70 \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 0.75x^2 - x^2 + 3x - 1.5x + 70 = 0 \]
\[ -0.25x^2 + 1.5x + 70 = 0 \]
Multiplying through by -4 to eliminate the decimal:
\[ x^2 - 6x - 280 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we need to factor the quadratic equation:
\[ x^2 - 6x - 280 = 0 \]
To factor, we look for two numbers that multiply to -280 and add to -6. The numbers are -20 and 14:
\[ (x - 20)(x + 14) = 0 \]
Thus, \( x = 20 \) or \( x = -14 \). Since \( x \) must be positive, we take \( x = 20 \).

### Step 8: Calculate the Number of Days for \( (x+10) \) Men
Now substituting \( x = 20 \):
- The number of men becomes:
\[ x + 10 = 20 + 10 = 30 \text{ men} \]
- The total work \( W \) is:
\[ W = (20 - 2) \times 20 = 18 \times 20 = 360 \]
- The number of days \( D \) for \( 30 \) men to finish the work is:
\[ D = \frac{W}{\text{Number of Men}} = \frac{360}{30} = 12 \text{ days} \]

### Final Answer
Thus, \( (x+10) \) men can finish the work in **12 days**.

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