Question

What is the sum of two numbers?

I. The bigger of these two numbers is $6$ more than the smaller number.

II. $40\%$ of the smaller number is equal to $30\%$ of the bigger number.

III. The ratio between half of the bigger number and one-third of the smaller number is $2:1$.

(A) II And III Are Sufficient.

(B) Only I And II Are Sufficient.

(C) Only II And III Are Sufficient.

(D) I And Either II Or III Are Sufficient.

• A. Option (D) is correct.
• B. Options (A), (b), and (C) are incorrect.

Answer 1

The correct option is A

Option (D) is correct.

Step 1. Solve the first given statement.

I. It is given that the bigger of these two numbers is $6$ more than the smaller number.

Consider the two numbers be $x$ and $y$ such that $y>x$.

$\Rightarrow y=x+6\dots \left(1\right)$

Step 2. Solve the second given statement.

II. It is given that $40\%$ of the smaller number is equal to $30\%$ of the bigger number.

$\begin{array}{rcl}& \Rightarrow & 40\%\mathrm{of}x=30\%\mathrm{of}y\\ & \Rightarrow & \frac{40}{100}·x=\frac{30}{100}·y\\ & \Rightarrow & \frac{x}{y}=\frac{3}{4}\end{array}$

$\Rightarrow 4x=3y\dots \left(2\right)$

Step 3. Solve the third given statement.

III. It is given that the ratio between half of the bigger number and one-third of the smaller number is $2:1$.

$\begin{array}{rcl}& \Rightarrow & \frac{1}{2}y:\frac{1}{3}x=2:1\\ & \Rightarrow & \frac{3y}{2x}=\frac{2}{1}\\ & \Rightarrow & \frac{x}{y}=\frac{3}{4}\end{array}$

$\Rightarrow 4x=3y\dots \left(3\right)$

Step 4: Determine the sum of two numbers.

Substitute the value of $y$ from $\left(1\right)$ into $\left(2\right)$.

$\begin{array}{rcl}4x& =& 3\left(x+6\right)\\ & \Rightarrow & 4x=3x+18\\ & \Rightarrow & 4x-3x=18\\ & \Rightarrow & x=18\end{array}$

Substitute the value of $x$ into $\left(1\right)$.

$\begin{array}{rcl}y& =& 18+6\\ & \Rightarrow & y=24\end{array}$

The sum of two numbers is $18+24=42$.

Hence, option (D) is correct.