Question

Find five rational numbers between (i) $\dfrac{7}{5}$ and $\dfrac{7}{6}$
(ii) $\dfrac{8}{5}$ and $\dfrac{8}{7}$

Answer 1

Hint: Here we find the equivalent fractions of both rational numbers so that their numerator or denominator become equal, then write the find rational number between them, repeat the same steps for (ii) part also.

Complete step-by-step answer:
(i) For finding rational numbers between $\dfrac{7}{5}$ and $\dfrac{7}{6}$
Here we have two rational numbers in which numerators of the same denominator are different. To make fractions with equal denominator, multiply the numerator and denominator of 7/5 by 6 and 7/6 by 5.
$\dfrac{{7 \times 6}}{{5 \times 6}} = \dfrac{{42}}{{30}}$
And
 $\dfrac{{7 \times 5}}{{6 \times 5}} = \dfrac{{35}}{{30}}$
These are equivalent fractions of the given rational numbers.
As we know that more the value of denominator less will be its value.
Required five rational numbers are
$\dfrac{{36}}{{30}},\dfrac{{37}}{{30}},\dfrac{{38}}{{30}},\dfrac{{39}}{{30}},\dfrac{{40}}{{30}}$
(ii) For finding rational numbers between $\dfrac{8}{5}$and $\dfrac{8}{7}$
Here we have two rational numbers in numerators are the same but denominators are different. To make fractions with equal denominator, multiply the numerator and denominator of 8/5 by 7 and 8/7 by 5.
$\dfrac{{8 \times 7}}{{5 \times 7}} = \dfrac{{56}}{{35}}$
And
 $\dfrac{{8 \times 5}}{{7 \times 5}} = \dfrac{{40}}{{35}}$
These are equivalent fractions of the given rational numbers.
As we know that for the same value of denominator more the value of numerator more be the value of fraction
Required five rational numbers are
$\dfrac{{41}}{{35}},\dfrac{{42}}{{35}},\dfrac{{43}}{{35}},\dfrac{{44}}{{35}},\dfrac{{45}}{{35}}$

Note: Always try to change the given fraction in like terms then you can easily compare the two rational numbers.
Alternatively, we can find the rational number between two rational numbers by taking their mean. Similarly we can find any number of rational numbers between the two given rational numbers.
Some important points: A rational number is represented in the form of p/q where q is not equal to 0. All integers are rational numbers i.e., 1, 2, 3, 4, 5 are rational numbers.