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Cross-multiplication is a crucial skill when working with proportions. It allows you to seamlessly transition from a ratio equation to a linear equation, which can then be solved more easily. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), you essentially equate the cross products: \( a \times d = b \times c \). This method is efficient because it eliminates the fractions by transforming them into whole numbers.

In the given exercise, the proportion \( \frac{2}{27} = \frac{r}{36} \) requires cross-multiplication. You multiply across the equal sign: \( 2 \times 36 = 27 \times r \), which simplifies to \( 72 = 27r \). This new equation, free of fractions, is now much easier to manipulate and solve. Cross-multiplication helps to simplify the process of solving for the unknown variable, providing a clear and efficient path towards the solution.

Once the proportion has been transformed into an equation via cross-multiplication, the next step is solving that equation to find the unknown value. Solving equations often involves isolating the variable on one side of the equation. In our example, we have \( 72 = 27r \). To solve this, you need to get \( r \) by itself on one side.

This involves dividing both sides of the equation by 27 to eliminate it from the right-hand side, giving you \( r = \frac{72}{27} \). Performing the same operation on both sides keeps the equation balanced and brings you closer to finding the value of the unknown. Solving equations is about performing operations to simplify and isolate variables, tapping into fundamental algebraic principles.

When you solve equations that involve proportions, you may end up with a fraction as part of your solution. Simplifying fractions is necessary to make your answer as clear and concise as possible. In the example where \( r = \frac{72}{27} \), you need to simplify this fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.

The GCD of 72 and 27 is 9. Dividing both the numerator and the denominator by 9 gives: \( \frac{72/9}{27/9} = \frac{8}{3} \). This results in a simpler and more understandable fraction for your final answer. Simplifying fractions not only makes the numbers smaller and easier to comprehend but also ensures consistency with standard mathematical practices. It is a vital skill for making your solutions as straightforward as possible.