Using concept of decimals, insert

(i) three rational numbers between 3 and 3.5

(ii) four rational numbers between $\dfrac{1}{4}$ and $\dfrac{2}{5}$.

(iii) five rational numbers between $1\dfrac{1}{2}$ and $1\dfrac{3}{4}$.

(i) We want three rational numbers between 3 and 3.5.

We know that,

Terminating decimals are rational numbers.

Let us take 3.1, 3.2, 3.3

Thus, we have :

⇒ 3 < 3.1 < 3.2 < 3.3 <3.5

Hence, three rational numbers between 3 and 3.5 are 3.1, 3.2 and 3.3.

(ii) Converting fraction in decimal form,

$\dfrac{1}{4}$ = 0.25

$\dfrac{2}{5}$ = 0.40

We know that,

Terminating decimals are rational numbers.

Let us take 0.28, 0.30, 0.32, 0.34

Thus, we have :

⇒ 0.25 < 0.28 < 0.30 < 0.32 < 0.34 < 0.40

Hence, four rational numbers between $\dfrac{1}{4} \text{ and } \dfrac{2}{5}$ = 0.28, 0.30, 0.32 and 0.34

(iii) We want five rational numbers between $1\dfrac{1}{2}$ and $1\dfrac{3}{4}$.

Numbers : $1\dfrac{1}{2} = \dfrac{3}{2} = 1.50$ and $1\dfrac{3}{4} = \dfrac{7}{4} = 1.75$

We know that,

Terminating decimals are rational numbers.

Let us take 1.61, 1.62, 1.63, 1.64, 1.65

Thus, we have :

⇒ 1.50 < 1.61 < 1.62 < 1.63 <1.64 < 1.65 <1.75

Hence, five rational numbers between $1\dfrac{1}{2} \text{ and } 1\dfrac{3}{4}$ = 1.61, 1.62, 1.63, 1.64 and 1.65.