Which of the following rational numbers does not lie between `(3)/(5) and (4)/(5)`?

Options

A. `(7)/(10)`
B. `(19)/(30)`
C. `(2)/(3)`
D. `(16)/(30)`

Answer

To determine which of the given rational numbers does not lie between \( \frac{3}{5} \) and \( \frac{4}{5} \), we will follow these steps:

### Step 1: Convert the fractions to a common denominator
The denominators of the fractions we are comparing are 5, 10, and 30. The least common multiple (LCM) of these denominators is 30.

### Step 2: Convert \( \frac{3}{5} \) and \( \frac{4}{5} \) to have a denominator of 30
- To convert \( \frac{3}{5} \) to a denominator of 30:
\[ \frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \]
- To convert \( \frac{4}{5} \) to a denominator of 30:
\[ \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \]

### Step 3: List the options and convert them to a denominator of 30
Now, we will convert the given options to have a denominator of 30:
1. \( \frac{7}{10} \):
\[ \frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} \]
2. \( \frac{19}{30} \) (already has a denominator of 30):
\[ \frac{19}{30} \]
3. \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \]
4. \( \frac{16}{30} \) (already has a denominator of 30):
\[ \frac{16}{30} \]

### Step 4: Compare the fractions
Now we have:
- \( \frac{3}{5} = \frac{18}{30} \)
- \( \frac{4}{5} = \frac{24}{30} \)

We can compare the converted options:
- \( \frac{21}{30} \) (lies between \( \frac{18}{30} \) and \( \frac{24}{30} \))
- \( \frac{19}{30} \) (lies between \( \frac{18}{30} \) and \( \frac{24}{30} \))
- \( \frac{20}{30} \) (lies between \( \frac{18}{30} \) and \( \frac{24}{30} \))
- \( \frac{16}{30} \) (does not lie between \( \frac{18}{30} \) and \( \frac{24}{30} \))

### Conclusion
The rational number that does not lie between \( \frac{3}{5} \) and \( \frac{4}{5} \) is:
\[ \frac{16}{30} \]