A takes 4 days more to complete a work than the time taken by`(A+B)` to do the same work and B takes 9 days more than the time taken by `(A+B)` to do the same work. In how many days `A+B` complete the work?

Answer

To solve the problem, we will use the information given about the time taken by A and B in relation to the time taken by A + B. Let's break it down step by step.

### Step 1: Define the variables
Let the time taken by A + B to complete the work be \( X \) days.

### Step 2: Express the time taken by A and B
According to the problem:
- A takes \( X + 4 \) days to complete the work.
- B takes \( X + 9 \) days to complete the work.

### Step 3: Write the work done equations
The work done by A in one day is \( \frac{1}{X + 4} \) and the work done by B in one day is \( \frac{1}{X + 9} \). The combined work done by A and B in one day is:
\[ \frac{1}{X + 4} + \frac{1}{X + 9} \]

### Step 4: Set up the equation
Since A and B together complete the work in \( X \) days, their combined work done in one day can also be expressed as \( \frac{1}{X} \). Therefore, we can set up the equation:
\[ \frac{1}{X + 4} + \frac{1}{X + 9} = \frac{1}{X} \]

### Step 5: Solve the equation
To solve the equation, we will find a common denominator:
\[ \frac{(X + 9) + (X + 4)}{(X + 4)(X + 9)} = \frac{1}{X} \]
This simplifies to:
\[ \frac{2X + 13}{(X + 4)(X + 9)} = \frac{1}{X} \]

Cross-multiplying gives:
\[ (2X + 13)X = (X + 4)(X + 9) \]

Expanding both sides:
\[ 2X^2 + 13X = X^2 + 13X + 36 \]

### Step 6: Rearranging the equation
Now, we can rearrange the equation:
\[ 2X^2 + 13X - X^2 - 13X - 36 = 0 \]
This simplifies to:
\[ X^2 - 36 = 0 \]

### Step 7: Solve for X
Factoring gives:
\[ (X - 6)(X + 6) = 0 \]
Thus, \( X = 6 \) or \( X = -6 \). Since time cannot be negative, we have:
\[ X = 6 \]

### Conclusion
Therefore, A + B can complete the work in **6 days**.

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