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Cross-multiplication is a powerful mathematical tool used to solve equations involving ratios or fractions by eliminating the fractions altogether. When you have a proportion, which is an equation that states two ratios are equivalent, you can use cross-multiplication to simplify and find the unknown.

In a proportion like \( \frac{a}{3} = \frac{16}{24} \), cross-multiplication involves multiplying across the equal sign diagonally:

• Multiply the numerator of the first fraction (\(a\)) by the denominator of the second fraction (24).
• Multiply the denominator of the first fraction (3) by the numerator of the second fraction (16).

This results in the equation \( 24a = 3 \times 16 \). The benefit of this method is that it turns a fraction-based equation into a simple linear equation that can be solved easily.

Ratios express a relationship between two quantities, showing how many times one value contains or is contained within another. Understanding ratios is crucial because they are used in various real-life applications such as cooking recipes, scale models, and financial analyses.

In the equation \( \frac{a}{3} = \frac{16}{24} \), the ratio \( \frac{a}{3} \) represents a relationship where 'a' is divided by 3, and \( \frac{16}{24} \) is a relationship between 16 and 24.

To determine if these two ratios form a true proportion, as they do here, one can use techniques like cross-multiplication to evaluate whether the two expressions are equivalent.
Working with ratios in this way helps us understand comparative sizes and scales.

After applying cross-multiplication to turn the proportion into a simple linear equation, the next step is to solve for the unknown variable. In the course of our example, this means isolating \(a\) on one side of the equation.

• First, recognize the equation as \( 24a = 48 \).
• To solve for \(a\), divide both sides by 24: \( a = \frac{48}{24} \).
• Simplify \(\frac{48}{24}\) by dividing both the numerator and the denominator by 24, yielding \( a = 2 \).

These steps demonstrate basic algebraic principles wherein you reverse the operations keeping the equation balanced to find an unknown variable. Solving equations like this relies on understanding the properties of equality and operations to systematically isolate the unknown.