Let's solve the problem step-by-step:
First, we need to determine the decimal equivalents of [tex]\(-\frac{4}{5}\)[/tex] and [tex]\(-\frac{3}{7}\)[/tex] so that we can compare.
1. Convert [tex]\(-\frac{4}{5}\)[/tex] to decimal:
[tex]\[
-\frac{4}{5} = -0.8
\][/tex]
2. Convert [tex]\(-\frac{3}{7}\)[/tex] to decimal:
[tex]\[
-\frac{3}{7} \approx -0.4286
\][/tex]
We are looking for rational numbers (fractions) that lie between [tex]\(-0.8\)[/tex] and [tex]\(-0.4286\)[/tex]. Let's examine each option:
### Option (a): [tex]\(-\frac{29}{15}\)[/tex] and [tex]\(-\frac{20}{15}\)[/tex]
1. Convert [tex]\(-\frac{29}{15}\)[/tex] to decimal:
[tex]\[
-\frac{29}{15} \approx -1.9333
\][/tex]
2. Convert [tex]\(-\frac{20}{15}\)[/tex] to decimal:
[tex]\[
-\frac{20}{15} \approx -1.3333
\][/tex]
Both [tex]\(-1.9333\)[/tex] and [tex]\(-1.3333\)[/tex] are less than [tex]\(-0.8\)[/tex]. Thus, neither of these two numbers lie between [tex]\(-0.8\)[/tex] and [tex]\(-0.4286\)[/tex].
### Option (b): [tex]\(-\frac{21}{35}\)[/tex] and [tex]\(-\frac{20}{35}\)[/tex]
1. Convert [tex]\(-\frac{21}{35}\)[/tex] to decimal:
[tex]\[
-\frac{21}{35} = -0.6
\][/tex]
2. Convert [tex]\(-\frac{20}{35}\)[/tex] to decimal:
[tex]\[
-\frac{20}{35} \approx -0.5714
\][/tex]
Both [tex]\(-0.6\)[/tex] and [tex]\(-0.5714\)[/tex] are between [tex]\(-0.8\)[/tex] and [tex]\(-0.4286\)[/tex]. Hence, these numbers satisfy the condition.
### Option (c): [tex]\(-\frac{28}{35}\)[/tex] and [tex]\(-\frac{25}{35}\)[/tex]
1. Convert [tex]\(-\frac{28}{35}\)[/tex] to decimal:
[tex]\[
-\frac{28}{35} \approx -0.8
\][/tex]
2. Convert [tex]\(-\frac{25}{35}\)[/tex] to decimal:
[tex]\[
-\frac{25}{35} \approx -0.7143
\][/tex]
Neither of these numbers are within the range [tex]\(-0.8\)[/tex] and [tex]\(-0.4286\)[/tex]. While [tex]\(-0.7143\)[/tex] could be considered in range, [tex]\(-0.8\)[/tex] is the lower boundary and does not strictly satisfy lying between the values.
### Option (d): [tex]\(-\frac{14}{35}\)[/tex] and [tex]\(-\frac{13}{35}\)[/tex]
1. Convert [tex]\(-\frac{14}{35}\)[/tex] to decimal:
[tex]\[
-\frac{14}{35} \approx -0.4
\][/tex]
2. Convert [tex]\(-\frac{13}{35}\)[/tex] to decimal:
[tex]\[
-\frac{13}{35} \approx -0.3714
\][/tex]
Both [tex]\(-0.4\)[/tex] and [tex]\(-0.3714\)[/tex] are greater than [tex]\(-0.4286\)[/tex]. Thus, neither of these numbers satisfy the condition.
### Conclusion:
The rational numbers in option (b) are the ones that lie between [tex]\(-\frac{4}{5}\)[/tex] and [tex]\(-\frac{3}{7}\)[/tex]:
[tex]\(\boxed{b}\)[/tex]
