Insert three rational numbers between `(1)/(3) and (4)/(5).`

Options

A. `(27)/(60), (17)/(60), (41)/(60)`
B. `(27)/(60), (17)/(30), (41)/(60)`
C. `(41)/(30), (17)/(30), (27)/(60)`
D. `(27)/(30),(17)/(30),(41)/(60)`

Answer

To find three rational numbers between \( \frac{1}{3} \) and \( \frac{4}{5} \), we can follow these steps:

### Step 1: Find a common denominator
The denominators of \( \frac{1}{3} \) and \( \frac{4}{5} \) are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.

### Step 2: Convert the fractions to have the same denominator
Convert \( \frac{1}{3} \) to a fraction with a denominator of 15:
\[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \]

Convert \( \frac{4}{5} \) to a fraction with a denominator of 15:
\[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \]

Now we have:
\[ \frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{4}{5} = \frac{12}{15} \]

### Step 3: Identify rational numbers between the two fractions
We need to find three rational numbers between \( \frac{5}{15} \) and \( \frac{12}{15} \). The rational numbers can be found by taking values between 5 and 12 in the numerator while keeping the denominator as 15.

### Step 4: Choose rational numbers
We can choose:
- \( \frac{6}{15} \)
- \( \frac{7}{15} \)
- \( \frac{8}{15} \)

These fractions are all between \( \frac{5}{15} \) and \( \frac{12}{15} \).

### Final Answer
The three rational numbers between \( \frac{1}{3} \) and \( \frac{4}{5} \) are:
\[ \frac{6}{15}, \frac{7}{15}, \frac{8}{15} \]

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