You are given three positive numbers such that

i) A is the sum of the first two numbers.

ii) B is the sum of the first two numbers taken away from the third number.

iii) C is the sum of all these numbers.

iv) $\frac{A}{B} = \frac{B}{C}$

Select the correct option from below:

Let the three positive numbers be $x$, $y$, and $z$.

From the given information, we can write:

Variable	Expression	Meaning
$A$	$x + y$	Sum of first two numbers
$B$	$z - (x + y)$	Third number minus sum of first two
$C$	$x + y + z$	Sum of all three numbers

Given that $\frac{A}{B} = \frac{B}{C}$, we can cross-multiply:

$A \times C = B^2$

Substituting the expressions:

$(x + y)(x + y + z) = [z - (x + y)]^2$

Let $s = x + y$ to simplify. The equation becomes:

$s(s + z) = (z - s)^2$

Expanding both sides:

$s^2 + sz = z^2 - 2sz + s^2$

The $s^2$ terms cancel out:

$sz = z^2 - 2sz$

Moving all terms with $s$ to one side:

$sz + 2sz = z^2$

$3sz = z^2$

Dividing both sides by $z$ (since $z$ is positive):

$3s = z$

Substituting back $s = x + y$:

$z = 3(x + y)$

This means the third number equals thrice the sum of the first two numbers.

Correct Option: 4