Find seven rational numbers between `1//6` and `5//21`.

Answer

To find seven rational numbers between \( \frac{1}{6} \) and \( \frac{5}{21} \), we can follow these steps:

### Step 1: Convert the fractions to have a common denominator.
To find rational numbers between the two fractions, we first need to convert them to have the same denominator.

The denominators are 6 and 21. We need to find the Least Common Multiple (LCM) of these two numbers.

- The LCM of 6 and 21 is 126.

### Step 2: Convert \( \frac{1}{6} \) to an equivalent fraction with a denominator of 126.
To convert \( \frac{1}{6} \) to a fraction with a denominator of 126, we multiply both the numerator and the denominator by 21:

\[ \frac{1}{6} = \frac{1 \times 21}{6 \times 21} = \frac{21}{126} \]

### Step 3: Convert \( \frac{5}{21} \) to an equivalent fraction with a denominator of 126.
To convert \( \frac{5}{21} \) to a fraction with a denominator of 126, we multiply both the numerator and the denominator by 6:

\[ \frac{5}{21} = \frac{5 \times 6}{21 \times 6} = \frac{30}{126} \]

### Step 4: Identify the range of rational numbers.
Now we have the two fractions as:

- \( \frac{1}{6} = \frac{21}{126} \)
- \( \frac{5}{21} = \frac{30}{126} \)

We need to find rational numbers between \( \frac{21}{126} \) and \( \frac{30}{126} \).

### Step 5: List the rational numbers between \( \frac{21}{126} \) and \( \frac{30}{126} \).
The rational numbers can be obtained by taking integers between 21 and 30 in the numerator while keeping the denominator as 126. The integers between 21 and 30 are 22, 23, 24, 25, 26, 27, and 28.

Thus, the seven rational numbers are:

\[ \frac{22}{126}, \frac{23}{126}, \frac{24}{126}, \frac{25}{126}, \frac{26}{126}, \frac{27}{126}, \frac{28}{126} \]

### Final Answer:
The seven rational numbers between \( \frac{1}{6} \) and \( \frac{5}{21} \) are:
\[ \frac{22}{126}, \frac{23}{126}, \frac{24}{126}, \frac{25}{126}, \frac{26}{126}, \frac{27}{126}, \frac{28}{126} \]

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