B would have taken 10 hours more than what A would have taken to complete a task if each of them worked alone. Working together they can complete the task in 12 hours. How many hours would B take to do 50% of the task?

Options

A. 30
B. 15
C. 20
D. 10

Answer

To solve the problem step by step, let's denote the time taken by A to complete the task alone as \( x \) hours. Then, according to the problem, B would take \( x + 10 \) hours to complete the same task alone.

### Step 1: Set up the equation for work done together
When A and B work together, they can complete the task in 12 hours. The work done by A in one hour is \( \frac{1}{x} \) and the work done by B in one hour is \( \frac{1}{x + 10} \). Therefore, the total work done by both A and B in one hour is:
\[ \frac{1}{x} + \frac{1}{x + 10} = \frac{1}{12} \]

### Step 2: Find a common denominator and solve the equation
To solve this equation, we find a common denominator:
\[ \frac{(x + 10) + x}{x(x + 10)} = \frac{1}{12} \]
This simplifies to:
\[ \frac{2x + 10}{x(x + 10)} = \frac{1}{12} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 12(2x + 10) = x(x + 10) \]
Expanding both sides results in:
\[ 24x + 120 = x^2 + 10x \]

### Step 4: Rearrange the equation
Rearranging the equation leads to:
\[ x^2 + 10x - 24x - 120 = 0 \]
which simplifies to:
\[ x^2 - 14x - 120 = 0 \]

### Step 5: Factor the quadratic equation
Next, we factor the quadratic:
\[ (x - 20)(x + 6) = 0 \]
This gives us two potential solutions:
\[ x = 20 \quad \text{or} \quad x = -6 \]
Since time cannot be negative, we take \( x = 20 \).

### Step 6: Calculate the time taken by B
Now, since A takes 20 hours, B takes:
\[ x + 10 = 20 + 10 = 30 \text{ hours} \]

### Step 7: Find the time taken by B to complete 50% of the task
To find out how long it would take B to complete 50% of the task, we calculate:
\[ \text{Time taken by B for 50% of the task} = \frac{30}{2} = 15 \text{ hours} \]

### Final Answer
Thus, B would take **15 hours** to complete 50% of the task.

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