A alone takes 4 more days to complete a work, in that time A and B can complete it together and B alone takes 16 more days to complete the same. In how many days they will do it together?

Options

A. 1. 6
B. 2. 8
C. 3. 12
D. 4. 10

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = the number of days A takes to complete the work alone.
- \( y \) = the number of days B takes to complete the work alone.

From the problem, we know:
- A takes 4 more days than the time they take together: \( x = t + 4 \)
- B takes 16 more days than the time they take together: \( y = t + 16 \)

### Step 2: Set Up the Equation
When A and B work together, their combined work rate is the sum of their individual work rates. The work rate can be expressed as:
- A's work rate = \( \frac{1}{x} \)
- B's work rate = \( \frac{1}{y} \)

Thus, their combined work rate when working together is:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{t} \]

### Step 3: Substitute the Values of \( x \) and \( y \)
Substituting \( x \) and \( y \) in terms of \( t \):
\[ \frac{1}{t + 4} + \frac{1}{t + 16} = \frac{1}{t} \]

### Step 4: Find a Common Denominator
To solve this equation, we can find a common denominator for the left side:
\[ \frac{(t + 16) + (t + 4)}{(t + 4)(t + 16)} = \frac{1}{t} \]
This simplifies to:
\[ \frac{2t + 20}{(t + 4)(t + 16)} = \frac{1}{t} \]

### Step 5: Cross-Multiply
Cross-multiplying gives:
\[ (2t + 20)t = (t + 4)(t + 16) \]

### Step 6: Expand Both Sides
Expanding both sides:
\[ 2t^2 + 20t = t^2 + 20t + 64 \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 2t^2 + 20t - t^2 - 20t - 64 = 0 \]
This simplifies to:
\[ t^2 - 64 = 0 \]

### Step 8: Solve for \( t \)
Factoring gives:
\[ (t - 8)(t + 8) = 0 \]
Thus, \( t = 8 \) (since time cannot be negative).

### Step 9: Conclusion
The time taken by A and B together to complete the work is \( t = 8 \) days.