Solving Proportions and Square Roots

This problem involves two main steps: first, solving a proportion to find the value of an unknown variable, and second, calculating the square root of an expression involving that variable.

Step 1: Understanding Proportions

A proportion is a statement that two ratios are equal. The given proportion is expressed as 15: 225 :: 35: x. This can be written mathematically as:

$ \frac{15}{225} = \frac{35}{x} $

Step 2: Calculating the Value of 'x'

To find the value of x, we can use cross-multiplication. Multiply the first term of the first ratio by the second term of the second ratio, and set it equal to the product of the second term of the first ratio and the first term of the second ratio.

$ 15 \times x = 225 \times 35 $

Now, we need to isolate x. Divide both sides of the equation by 15:

$ x = \frac{225 \times 35}{15} $

We can simplify this calculation. Notice that 225 divided by 15 is 15 (since $15 \times 15 = 225$).

$ x = 15 \times 35 $

Performing the multiplication:

$ x = 525 $

Step 3: Calculating the Square Root Expression

The second part of the question asks for the value of $\sqrt{x + 100}$. Now that we know x = 525, we can substitute this value into the expression:

$ \sqrt{x + 100} = \sqrt{525 + 100} $

Add the numbers inside the square root:

$ \sqrt{525 + 100} = \sqrt{625} $

Finally, calculate the square root of 625:

$ \sqrt{625} = 25 $

Therefore, the value of $\sqrt{x + 100}$ is 25.