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Understanding how to solve proportions is a fundamental skill in algebra. A proportion is an equation that equates two ratios or fractions. To solve for an unknown variable in a proportion, cross-multiplication is often the method of choice.

Let's consider the given exercise: \(\frac{5.6}{12.7} = \frac{4.9}{x}\). This is a proportion because two ratios are set equal to each other. One of the ratios contains the unknown variable \(x\), which we aim to find. By cross-multiplying, we are creating a regular algebraic equation without fractions, which simplifies the process of solving for \(x\).

The fundamental idea is to multiply the outer terms together, and the inner terms together, which yields: \(5.6 * x = 12.7 * 4.9\). This equation can then be solved for \(x\) using standard algebraic techniques.

Working with fractions is another key aspect of solving proportions. When presented with an equation with fractions, it's important to know how to manipulate these fractions to solve for the unknown variable.

Operations on fractions typically involve finding a common denominator, simplifying, and sometimes multiplying or dividing. However, when dealing with proportions, cross-multiplication allows you to avoid these steps by converting the proportion into a simpler equation.

In our example \(\frac{5.6}{12.7} = \frac{4.9}{x}\), we cross-multiply to eliminate the fractions. Understanding how to carry out this multiplication correctly is essential, as it sets the stage for the subsequent steps in solving the equation.

After transforming our proportion into an algebraic equation using cross-multiplication, we now apply principles of algebra to solve for our unknown variable, \(x\). In algebra, we often manipulate equations to isolate the variable of interest on one side.

For the given problem, once we have the equation \(5.6 * x = 62.23\) from cross-multiplication, we need to isolate \(x\). This is accomplished by dividing both sides of the equation by \(5.6\), resulting in \(x = \frac{62.23}{5.6}\). The division operation completes the process of solving the equation, and we compute the value of \(x\) to be 11.11.

The stepwise approach of transforming a proportion into an algebraic equation and then isolating and solving for the variable is the heart of solving such problems in algebra.