Amy completes a work in t day and with Bob. Amy takes n day for the same work. If Bobo can completes twice the same work inside of 16 day, which of the following gives the correct relation between t and n ?

Options

A. `n lt (8t)/(8 +t)`
B. `n lt (8+t)/(8t)`
C. `n gt (t)/(8t +1)`
D. ` n gt (t)/(8 + t)`

Answer

To solve the problem step by step, we will analyze the information given and derive the relationship between \( t \) and \( n \).

### Step 1: Determine the work done by Amy and Bob
Let:
- Work done by Amy in 1 day = \( A \)
- Work done by Bob in 1 day = \( B \)

From the problem, we know:
- Amy completes the work in \( t \) days, so:
\[ A = \frac{1}{t} \]
- Amy takes \( n \) days to complete the same work with Bob, so:
\[ A + B = \frac{1}{n} \]

### Step 2: Express Bob's work in terms of \( t \) and \( n \)
Substituting \( A \) into the equation:
\[ \frac{1}{t} + B = \frac{1}{n} \]
Rearranging gives:
\[ B = \frac{1}{n} - \frac{1}{t} \]

### Step 3: Determine Bob's work rate
It is given that Bob can complete twice the work in less than 16 days. Therefore, the work done by Bob in 1 day can be expressed as:
\[ B > \frac{2 \text{ works}}{16 \text{ days}} = \frac{1}{8} \]

### Step 4: Set up the inequality
From the previous step, we have:
\[ \frac{1}{n} - \frac{1}{t} > \frac{1}{8} \]

### Step 5: Solve the inequality for \( n \)
Rearranging the inequality:
\[ \frac{1}{n} > \frac{1}{8} + \frac{1}{t} \]
Finding a common denominator (which is \( 8t \)):
\[ \frac{1}{n} > \frac{t + 8}{8t} \]

### Step 6: Invert the inequality
Taking the reciprocal (and remembering to flip the inequality):
\[ n < \frac{8t}{t + 8} \]

### Conclusion
Thus, the correct relation between \( t \) and \( n \) is:
\[ n < \frac{8t}{t + 8} \]