Introduction to Ratios, Proportions, and Fraction Equivalence
In mathematics, a ratio is a comparison of two numbers indicating how many times one value contains another. When two ratios are set equal to each other, they form a proportion. Proportions represent equivalent fractions. Formally, a proportion is written as:
\[\frac{A}{B} = \frac{C}{D}\]
Where \(B \ne 0\) and \(D \ne 0\). In this mathematical statement, \(A\) and \(D\) are called the extremes, while \(B\) and \(C\) are called the means. The fundamental property of proportions states that the product of the extremes equals the product of the means. This leads directly to the algebraic technique known as cross multiplication.
Cross multiplication is an efficient way to clear denominators from a fractional equation. By multiplying both sides of the equation by the product of the denominators (known as the least common multiple), the fractional terms are converted into a simple linear equation. This makes cross multiplication a vital tool for solving algebraic equations containing unknowns.