`(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, we will follow these steps:

### Step 1: Set up the equations based on the information given.
We know that `(x - 2)` men can do a piece of work in `x` days. Therefore, the total work (W) can be expressed as:
\[ W = (x - 2) \times x \]

### Step 2: Express the work done by `(x + 7)` men in `(x - 10)` days.
According to the problem, `(x + 7)` men can do `75%` of the same work in `(x - 10)` days. Thus, the work done by `(x + 7)` men can be expressed as:
\[ W_{75\%} = (x + 7) \times (x - 10) \]
Since this is `75%` of the total work, we can write:
\[ W_{75\%} = 0.75 \times W = 0.75 \times (x - 2) \times x \]

### Step 3: Set the two expressions for work equal to each other.
Now we can set the two expressions for work equal to each other:
\[ (x + 7)(x - 10) = 0.75 \times (x - 2) \times x \]

### Step 4: Expand both sides of the equation.
Expanding the left side:
\[ (x + 7)(x - 10) = x^2 - 10x + 7x - 70 = x^2 - 3x - 70 \]

Expanding the right side:
\[ 0.75 \times (x - 2) \times x = 0.75(x^2 - 2x) = 0.75x^2 - 1.5x \]

### Step 5: Set the equation to zero.
Now we can set the equation:
\[ x^2 - 3x - 70 = 0.75x^2 - 1.5x \]

Rearranging gives:
\[ x^2 - 0.75x^2 - 3x + 1.5x - 70 = 0 \]
\[ 0.25x^2 - 1.5x - 70 = 0 \]

### Step 6: Multiply through by 4 to eliminate the decimal.
Multiplying the entire equation by 4:
\[ x^2 - 6x - 280 = 0 \]

### Step 7: Factor the quadratic equation.
We can factor the quadratic:
\[ (x - 20)(x + 14) = 0 \]

### Step 8: Solve for x.
Setting each factor to zero gives:
1. \( x - 20 = 0 \) → \( x = 20 \)
2. \( x + 14 = 0 \) → \( x = -14 \) (not valid since x must be positive)

Thus, \( x = 20 \).

### Step 9: Find the number of men.
Now, we need to find the number of men when \( x + 10 \):
\[ x + 10 = 20 + 10 = 30 \text{ men} \]

### Step 10: Calculate the total work.
Using the total work formula:
\[ W = (x - 2) \times x = (20 - 2) \times 20 = 18 \times 20 = 360 \text{ units} \]

### Step 11: Calculate the number of days for 30 men to finish the work.
Using the formula for the time taken:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Number of Men}} = \frac{360}{30} = 12 \text{ days} \]

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

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