To find a rational number (fraction in simplest form) between \( 6 \) and \( 6 \frac{1}{4} \), we can first convert \( 6 \frac{1}{4} \) into an improper fraction.
- Convert \( 6 \frac{1}{4} \): \[ 6 \frac{1}{4} = 6 + \frac{1}{4} = \frac{6 \times 4}{4} + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4} \]
Now we need to find a rational number between \( 6 \) (which can be written as \( \frac{24}{4} \)) and \( \frac{25}{4} \).
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The simplest way to find a rational number between \( \frac{24}{4} \) and \( \frac{25}{4} \) is to take the average of the two fractions: \[ \text{Average} = \frac{\frac{24}{4} + \frac{25}{4}}{2} \] Combine the fractions: \[ = \frac{\frac{24 + 25}{4}}{2} = \frac{\frac{49}{4}}{2} = \frac{49}{4} \times \frac{1}{2} = \frac{49}{8} \]
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Now, \( \frac{49}{8} \) is a rational number between \( 6 \) and \( 6\frac{1}{4} \). To confirm:
- Convert \( 6 \) to eighths: \[ 6 = \frac{48}{8} \]
- Convert \( 6\frac{1}{4} \) to eighths: \[ 6\frac{1}{4} = \frac{25}{4} = \frac{50}{8} \]
Since \( \frac{48}{8} < \frac{49}{8} < \frac{50}{8} \), it confirms that \( \frac{49}{8} \) is indeed between \( 6 \) and \( 6 \frac{1}{4} \).
- Lastly, check if \( \frac{49}{8} \) is in simplest form:
- The greatest common divisor (GCD) of 49 and 8 is 1 (since 49 is \( 7^2 \) and 8 is \( 2^3 \)).
- Thus \( \frac{49}{8} \) is already in simplest form.
Therefore, the rational number between \( 6 \) and \( 6 \frac{1}{4} \) is: \[ \frac{49}{8} \]