When x is added to each of the numbers 7, 11, 18 and 23, then the number so obtained are in proportion. What is the mean proportional between the (x – 1) and (2x – 10)?

The correct answer is Option 1: 48.

We are given four numbers: 7, 11, 18, and 23. When a number $x$ is added to each of these numbers, the resulting numbers are in proportion.

This means:

$\frac{7+x}{11+x}=\frac{18+x}{23+x}$

Cross-multiplying to solve for $x$:

$(7+x)(23+x)=(18+x)(11+x)$

Expanding both sides:

$161+7x+23x+x^2=198+18x+11x+x^2$

$161+30x+x^2=198+29x+x^2$

Subtracting $x^2$ from both sides:

$161+30x=198+29x$

Rearranging terms to solve for $x$:

$30x-29x=198-161$

$x=37$

Now, we need to find the mean proportional between $(x-1)$ and $(2x-10)$.

First, calculate the values of these two expressions for $x=37$:

$x-1=37-1=36$

$2x-10=2\left(37\right)-10=74-10=64$

The mean proportional between two numbers $a$ and $b$ is given by $\sqrt{a\times b}$.

So, the mean proportional is:

$\sqrt{36\times 64}=\sqrt{36}\times \sqrt{64}=6\times 8=48$

Thus, the mean proportional is 48.