Find five rational numbers between `(3)/(4) "and" (7)/(8)`.

Answer

To find five rational numbers between \( \frac{3}{4} \) and \( \frac{7}{8} \), we will follow these steps:

### Step 1: Convert the fractions to have a common denominator
The first step is to find a common denominator for \( \frac{3}{4} \) and \( \frac{7}{8} \). The denominators are 4 and 8. The least common multiple (LCM) of 4 and 8 is 8.

- Convert \( \frac{3}{4} \) to have a denominator of 8:
\[ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \]

Now we have:
- \( \frac{6}{8} \) and \( \frac{7}{8} \)

### Step 2: Identify the range between the two fractions
Now we need to find rational numbers between \( \frac{6}{8} \) and \( \frac{7}{8} \).

### Step 3: Multiply to find more rational numbers
To find five rational numbers between \( \frac{6}{8} \) and \( \frac{7}{8} \), we can multiply both fractions by a number greater than 5. Let's multiply by 6.

- Multiply \( \frac{6}{8} \) by 6:
\[ \frac{6}{8} \times \frac{6}{6} = \frac{36}{48} \]

- Multiply \( \frac{7}{8} \) by 6:
\[ \frac{7}{8} \times \frac{6}{6} = \frac{42}{48} \]

Now we have:
- \( \frac{36}{48} \) and \( \frac{42}{48} \)

### Step 4: List the rational numbers between \( \frac{36}{48} \) and \( \frac{42}{48} \)
The rational numbers between \( \frac{36}{48} \) and \( \frac{42}{48} \) can be found by simply listing the fractions with numerators from 37 to 41:
- \( \frac{37}{48} \)
- \( \frac{38}{48} \)
- \( \frac{39}{48} \)
- \( \frac{40}{48} \)
- \( \frac{41}{48} \)

### Final Answer
The five rational numbers between \( \frac{3}{4} \) and \( \frac{7}{8} \) are:
- \( \frac{37}{48} \)
- \( \frac{38}{48} \)
- \( \frac{39}{48} \)
- \( \frac{40}{48} \)
- \( \frac{41}{48} \)

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