Which of the following lists those integer value of D for which the fraction `2/D` lies between `1/5 and 1/3`?

Options

A. 4 only
B. 3,4 and 5
C. 8 only
D. 7, 8, and 9

Answer

To solve the problem of finding the integer values of \( D \) for which the fraction \( \frac{2}{D} \) lies between \( \frac{1}{5} \) and \( \frac{1}{3} \), we can follow these steps:

### Step 1: Set up the inequality
We need to express the condition mathematically:
\[ \frac{1}{5} < \frac{2}{D} < \frac{1}{3} \]

### Step 2: Break it into two inequalities
We can break this into two separate inequalities:
1. \( \frac{1}{5} < \frac{2}{D} \)
2. \( \frac{2}{D} < \frac{1}{3} \)

### Step 3: Solve the first inequality
Starting with the first inequality:
\[ \frac{1}{5} < \frac{2}{D} \]
To eliminate the fraction, we can cross-multiply (noting that \( D \) must be positive):
\[ 1 \cdot D < 2 \cdot 5 \]
This simplifies to:
\[ D < 10 \]

### Step 4: Solve the second inequality
Now, we solve the second inequality:
\[ \frac{2}{D} < \frac{1}{3} \]
Again, we cross-multiply:
\[ 2 \cdot 3 < 1 \cdot D \]
This simplifies to:
\[ 6 < D \quad \text{or} \quad D > 6 \]

### Step 5: Combine the results
Now we combine the results from both inequalities:
\[ 6 < D < 10 \]

### Step 6: Identify integer values
The integer values of \( D \) that satisfy this inequality are:
\[ D = 7, 8, 9 \]

### Final Answer
Thus, the integer values of \( D \) for which \( \frac{2}{D} \) lies between \( \frac{1}{5} \) and \( \frac{1}{3} \) are:
\[ \{7, 8, 9\} \]