Why Does Cross Multiplying Work to Compare Fractions?
Comparing fractions is a fundamental skill in mathematics, and cross multiplying is one of the most common methods used to do so. But have you ever wondered why cross multiplying works? In this article, we will explore the concept of cross multiplying and understand the underlying principles that make it an effective tool for comparing fractions.
Understanding the Concept of Fractions
Before we delve into the mechanics of cross multiplying, it’s essential to have a clear understanding of what fractions represent. A fraction consists of two numbers: the numerator and the denominator. The numerator represents the number of parts we have, while the denominator represents the total number of parts in the whole. For example, the fraction 3/4 means we have three parts out of a total of four parts.
The Need for Cross Multiplication
When comparing two fractions, we want to determine which one is larger or smaller. However, comparing fractions with different denominators can be challenging. This is where cross multiplying comes into play. Cross multiplying allows us to equate the two fractions by multiplying the numerator of one fraction with the denominator of the other and vice versa. By doing so, we can create equivalent fractions with the same denominator, making it easier to compare them.
The Mathematical Explanation
The reason cross multiplying works lies in the properties of fractions and the concept of proportionality. When we cross multiply, we are essentially setting up a proportion. Let’s consider two fractions, a/b and c/d. To compare them, we cross multiply as follows:
a/b = c/d
ad = bc
If ad = bc, then the two fractions are equal. If ad > bc, then a/b is greater than c/d. Conversely, if ad < bc, then a/b is smaller than c/d.
The Underlying Principle
The underlying principle behind cross multiplying is that multiplying the numerator of one fraction by the denominator of the other and vice versa preserves the relative value of the fractions. This is because we are essentially multiplying the parts we have (numerator) by the total parts in the whole (denominator) for both fractions. By doing so, we create equivalent fractions with the same denominator, allowing us to compare their values more easily.
Conclusion
In conclusion, cross multiplying works to compare fractions because it leverages the properties of fractions and the concept of proportionality. By setting up a proportion and multiplying the numerators and denominators, we can determine the relative values of two fractions with different denominators. Understanding the underlying principles behind cross multiplying can help us appreciate its effectiveness and apply it confidently in various mathematical contexts.
