Introduction:
In this question, we are given three numbers - 100, x, and 256. We need to find the value of x. To solve this problem, we can use the concept of ratios and proportions.

Understanding Ratios:
A ratio is a comparison of two or more numbers. It represents the relationship or proportion between them. Ratios can be written in different formats, such as using a colon (:) or as a fraction. For example, the ratio of 2 to 3 can be written as 2:3 or as a fraction 2/3.

Understanding Proportions:
A proportion is an equation that states that two ratios are equal. In other words, if we have two equal ratios, we can form a proportion. Proportions are useful in finding the missing values in a ratio.

Solving the Problem:
To find the value of x, we can set up a proportion using the given numbers. We know that the ratio of 100 to x is equal to the ratio of x to 256. Mathematically, we can represent this as:

100/x = x/256

Cross-Multiplication:
To solve this proportion, we can use the concept of cross-multiplication. Cross-multiplication involves multiplying the numerator of one ratio with the denominator of the other ratio. In this case, we can multiply 100 with 256 and x with x:

100 * x = 256 * x

Simplifying the Equation:
Now, we can simplify the equation by canceling out the common factor of x on both sides:

100 = 256

Conclusion:
After simplifying the equation, we can see that we are left with the equation 100 = 256, which is not possible. This means that there is no value of x that can satisfy the given proportion. Therefore, there is no solution for x in this case.

Summary:
- We were given three numbers - 100, x, and 256 and we needed to find the value of x.
- We used the concept of ratios and proportions to solve the problem.
- Setting up a proportion, we formed the equation 100/x = x/256.
- Using cross-multiplication, we simplified the equation to 100 = 256.
- Since the equation is not possible, there is no solution for x in this case.