When x is added to each of 13, 19, 16 and 23, then the numbers so obtained, in this order, are in proportion. Then, if $\mathrm{ 5}x:y::y:(8x-4)$, and y > 0, what is the value of y?613

The correct option is 30.

Step 1: Solve for x

We are given that when $x$ is added to each of the numbers 13, 19, 16, and 23, the resulting numbers are in proportion in that order. This gives us the following relationship:

$\frac{13+x}{19+x}=\frac{16+x}{23+x}$

To solve for $x$, we cross-multiply:

$(13+x)(23+x)=(16+x)(19+x)$

Expanding both sides of the equation:

$299+13x+23x+x^2=304+16x+19x+x^2$

Simplify by combining like terms:

Subtract $x^2$ from both sides:

$299+36x=304+35x$

Subtract $35x$ and 299 from both sides to isolate $x$:

$x=304-299$

$x=5$

Step 2: Solve for y using the proportion

We are given the second proportion:

$5x:y::y:(8x-4)$

This can be written in fraction form as:

$\frac{5x}{y}=\frac{y}{8x-4}$

Cross-multiplying gives:

$y^2=5x(8x-4)$

Substitute the value $x=5$ into the equation:

$y^2=5\left(5\right)\cdot \left(8\right(5)-4)$

$y^2=25\cdot (40-4)$

$y^2=25\cdot 36$

Taking the square root on both sides (and since $y>0$):

$y=\sqrt{25\cdot 36}$

$y=5\cdot 6$

$y=30$

Therefore, the value of $y$ is 30.