Divide ₹ 782 among P, Q and R in the ratio $\dfrac{1}{2} : \dfrac{2}{3} : \dfrac{3}{4}$.

Given,

P : Q : R = $\dfrac{1}{2} : \dfrac{2}{3} : \dfrac{3}{4}$.

Multiply each ratio by 12 (LCM of denominators) to clear fractions :

= $\dfrac{1}{2} \times 12 : \dfrac{2}{3} \times 12 : \dfrac{3}{4} \times 12$

= 6 : 4 : 9.

Let P = 6a and Q = 8a and R = 9a.

To find P's part,

$\Rightarrow \dfrac{6a}{6a + 8a + 9a} \times 782 \\[1em] \Rightarrow \dfrac{6a}{23a} \times 782 \\[1em] \Rightarrow \dfrac{6}{23} \times 782 \\[1em] \Rightarrow 204$

To find Q's part,

$\Rightarrow \dfrac{8a}{6a + 8a + 9a} \times 782 \\[1em] \Rightarrow \dfrac{8a}{23a} \times 782 \\[1em] \Rightarrow \dfrac{8}{23} \times 782 \\[1em] \Rightarrow 272$

To find Q's part,

$\Rightarrow \dfrac{9a}{6a + 8a + 9a} \times 782 \\[1em] \Rightarrow \dfrac{9a}{23a} \times 782 \\[1em] \Rightarrow \dfrac{9}{23} \times 782 \\[1em] \Rightarrow 306$

Hence, ₹ 782 can be divided into P = ₹ 204, Q = ₹ 272 and R = ₹ 306.