If Rs. 782 is divided into three parts, proportional to
$$\begin{aligned} \frac{1}{2}:\frac{2}{3}:\frac{3}{4} \end{aligned}$$

find the first part.

Answer: Option D

Explanation:

$$\begin{aligned} a:b = 2:3 \\ b:c = 5:8 = (5*\frac{3}{5} : 8*\frac{3}{5}) \\ = 3:\frac{24}{5} \\ a:b:c = 2:3:\frac{24}{5} \\ = 10:15:24 \\ b = 98*\frac{15}{49} \\ = 30 \end{aligned}$$

Answer: Option B

Explanation:

$$\begin{aligned} \frac{1}{2}:\frac{2}{3}:\frac{3}{4} \\ = 6:8:9\\ \text{First part =} 782 * \frac{6}{23} \\ = 204 \end{aligned}$$

Answer: Option B

Explanation:

Dog : Hare = (3*3) leaps of hare : 5 leaps of hare
= 9 : 5

Answer: Option B

Explanation:

$$\begin{aligned} A:B = 2:3 \\ B:C = 4:5 = (4*\frac{3}{4} : 5*\frac{3}{4}) \\ = 3:\frac{15}{4} \\ C:D = 6:7 = (6*\frac{15}{24} : 7*\frac{15}{24}) \\ = \frac{15}{4}:\frac{35}{8} \\ A:B:C:D = 2:3:\frac{15}{4}:\frac{35}{8} \\ = 16:24:30:35 = 8:12:9 \end{aligned}$$

Answer: Option D

Explanation:

Let the first number is A and second number is B.

As per question

$$\begin{aligned} \frac{40}{100}A = \frac{2}{3}B \\ \frac{A}{B} = \frac{2}{3}*\frac{100}{40} \\ \frac{A}{B} = \frac{5}{3} \\ => A:B = 5:3 \end{aligned}$$