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 C

Explanation:

let x = 1k and y = 3k, so
\begin{aligned}
= \frac{7(k)+3(3k)}{2(k)+1(3k)} \\
= \frac{16k}{5k} \\
= 16:5
\end{aligned}

Answer: Option D

Explanation:

From question, lets assume that two numbers are 3x and 5x.

\begin{aligned}
\text{then, }\frac{3x - 9}{ 5x - 9} = \frac{12}{23}\\
=> \text{23(3x - 9) = 12(5x - 9)}\\
=> 9x = 99 \\
=> x = 11
\end{aligned}
So numbers are 3(11) = 33, 5(11) = 55.
Smaller number is 33.

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}

Answer: Option B

Explanation:

Let the numbers be 3x, 2x and 5x.
Then,
9x + 4x + 25x =1862
⇒ 38x = 1862
⇒ x = 49 ⇒ x = 7.
middle number = 2x = 14

Answer: Option A

Explanation:

\begin{aligned}
a:b = 2:3 \\
b:c = 4:3 = (4*\frac{3}{4} : 3*\frac{3}{4}) \\
= 3:\frac{9}{4} \\
a:b:c = 2:3:\frac{9}{4} \\
= 8:12:9
\end{aligned}