Which of the following is not a rational number between `(3)/(4) and (1)/(2)` ?

Options

A. `(9)/(16)`
B. `(13)/(16)`
C. `(10)/(16)`
D. `(11)/(16)`

Answer

To determine which of the following is not a rational number between \( \frac{3}{4} \) and \( \frac{1}{2} \), we can follow these steps:

### Step 1: Convert the fractions to have a common denominator
We need to convert \( \frac{3}{4} \) and \( \frac{1}{2} \) to have a common denominator to easily compare them. The least common multiple of 4 and 2 is 4.

- \( \frac{3}{4} \) remains \( \frac{3}{4} \).
- Convert \( \frac{1}{2} \) to have a denominator of 4:
\[ \frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \]

### Step 2: Identify the range of rational numbers
Now we have:
- \( \frac{3}{4} = \frac{3}{4} \)
- \( \frac{1}{2} = \frac{2}{4} \)

The range of rational numbers between \( \frac{1}{2} \) and \( \frac{3}{4} \) is between \( \frac{2}{4} \) and \( \frac{3}{4} \).

### Step 3: Find rational numbers in the range
We can express these fractions with a common denominator of 16 to find rational numbers between them:
- Convert \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16} \]
- Convert \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{1 \times 8}{2 \times 8} = \frac{8}{16} \]

### Step 4: Identify rational numbers between \( \frac{8}{16} \) and \( \frac{12}{16} \)
Now, we need to find rational numbers between \( \frac{8}{16} \) and \( \frac{12}{16} \):
- The rational numbers can be \( \frac{9}{16}, \frac{10}{16}, \frac{11}{16} \).

### Step 5: Check the options
Now we check the given options to see which one does not fall between \( \frac{8}{16} \) and \( \frac{12}{16} \):
- \( \frac{9}{16} \) (between)
- \( \frac{10}{16} \) (between)
- \( \frac{11}{16} \) (between)
- \( \frac{13}{16} \) (not between)

### Conclusion
Thus, the rational number that is not between \( \frac{3}{4} \) and \( \frac{1}{2} \) is \( \frac{13}{16} \).

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