To find two rational numbers between [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{5}{7} \)[/tex], we will use a clear and logical approach to locate values that fall within this interval. Here is a detailed, step-by-step solution:
### Step 1: Understand the given fractions
The two given fractions are:
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
[tex]\[ \frac{5}{7} \approx 0.714285714 \][/tex]
### Step 2: Identify the mid-point
One effective way to find a rational number between two fractions is to find their arithmetic average (i.e., the midpoint). To find the midpoint between [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{5}{7} \)[/tex], we use the formula for the average of two numbers:
[tex]\[ \text{Midpoint} = \frac{\frac{3}{5} + \frac{5}{7}}{2} \][/tex]
#### Convert fractions to decimals
[tex]\[ \frac{3}{5} = 0.6 \][/tex]
[tex]\[ \frac{5}{7} \approx 0.714285714 \][/tex]
#### Add the fractions
[tex]\[ 0.6 + 0.714285714 \approx 1.314285714 \][/tex]
#### Divide by 2
[tex]\[ \frac{1.314285714}{2} \approx 0.657142857 \][/tex]
So, one rational number between [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{5}{7} \)[/tex] is approximately:
[tex]\[ 0.657142857 \][/tex]
### Step 3: Find another rational number between [tex]\(\frac{3}{5}\)[/tex] and the first rational number
To find another rational number between [tex]\(\frac{3}{5}\)[/tex] and [tex]\( 0.657142857 \)[/tex], we use the same approach: calculate the midpoint between [tex]\(\frac{3}{5}\)[/tex] and [tex]\( 0.657142857 \)[/tex].
#### Add the fractions again
[tex]\[ 0.6 + 0.657142857 = 1.257142857 \][/tex]
#### Divide by 2
[tex]\[ \frac{1.257142857}{2} \approx 0.628571429 \][/tex]
Hence, another rational number between [tex]\(\frac{3}{5}\)[/tex] and [tex]\( \frac{5}{7} \)[/tex] is approximately:
[tex]\[ 0.628571429 \][/tex]
### Conclusion
The two rational numbers between [tex]\( \frac{3}{5} \)[/tex] and [tex]\( \frac{5}{7} \)[/tex] are:
[tex]\[ 0.657142857 \][/tex]
and
[tex]\[ 0.628571429 \][/tex]
