Sure, let's find a rational number between [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex].
### Step-by-Step Solution
Step 1: Convert the fractions to a common denominator.
First, let's convert both fractions, [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex], to have a common denominator. To do this, we can find the least common multiple (LCM) of the denominators 7 and 10.
The LCM of 7 and 10 is 70.
Now, let's rewrite the fractions with this common denominator:
- [tex]\(\frac{3}{7}\)[/tex]:
Convert to a denominator of 70:
[tex]\[
\frac{3}{7} = \frac{3 \times 10}{7 \times 10} = \frac{30}{70}
\][/tex]
- [tex]\(\frac{6}{10}\)[/tex]:
Convert to a denominator of 70:
[tex]\[
\frac{6}{10} = \frac{6 \times 7}{10 \times 7} = \frac{42}{70}
\][/tex]
So, [tex]\(\frac{3}{7} = \frac{30}{70}\)[/tex] and [tex]\(\frac{6}{10} = \frac{42}{70}\)[/tex].
Step 2: Check for Rational Numbers Between the Two Fractions
We need to find a rational number that lies between [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{42}{70}\)[/tex].
Step 3: Calculate the Average as a Middle Fraction
One reliable way to find a rational number between two fractions is to take their average.
Calculate the average of [tex]\(\frac{30}{70}\)[/tex] and [tex]\(\frac{42}{70}\)[/tex]:
[tex]\[
\text{Average} = \frac{\frac{30}{70} + \frac{42}{70}}{2} = \frac{30 + 42}{2 \times 70} = \frac{72}{140}
\][/tex]
Step 4: Simplify the Resulting Fraction
Now, simplify [tex]\(\frac{72}{140}\)[/tex]:
- The greatest common divisor (GCD) of 72 and 140 is 2.
Simplify by dividing the numerator and the denominator by their GCD:
[tex]\[
\frac{72 \div 2}{140 \div 2} = \frac{36}{70} = \frac{18}{35}
\][/tex]
Final Result:
The fraction [tex]\(\frac{18}{35}\)[/tex] is a rational number between [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex].
So, a rational number between [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{6}{10}\)[/tex] is [tex]\(\frac{18}{35}\)[/tex].
