Cross multiplication is a common method for comparing fractions. It involves multiplying the numerator of one fraction by the denominator of the other, and vice versa. But why does this technique work? This article explores the underlying mathematical principles that make cross multiplication a valid method for comparing fractions.
The Principle Behind Cross Multiplication
Cross multiplication is fundamentally based on the concept of equivalent fractions. Two fractions are equivalent if they represent the same value, even if they have different numerators and denominators. For instance, 1/2 and 2/4 are equivalent because they both represent half. To create equivalent fractions, you multiply both the numerator and the denominator by the same non-zero number.
When we cross multiply two fractions, say a/b and c/d, we are essentially finding a common denominator (b*d) and scaling both fractions to have that denominator.
- a/b becomes (ad)/(bd)
- c/d becomes (cb)/(bd)
Now, with a common denominator, comparing the two fractions becomes as simple as comparing their numerators: (ad) and (cb). If (ad) > (cb), then a/b > c/d. If (ad) < (cb), then a/b < c/d. And if (ad) = (cb), then a/b = c/d.
Cross Multiplication in Action: Comparing Fractions
Let’s illustrate this with an example: Compare 3/4 and 5/7.
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Cross Multiply:
- 3 * 7 = 21
- 4 * 5 = 20
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Compare the Products: Since 21 > 20, we conclude that 3/4 > 5/7.
This result aligns with finding a common denominator:
- 3/4 = (37)/(47) = 21/28
- 5/7 = (54)/(74) = 20/28
As you can see, 21/28 (representing 3/4) is greater than 20/28 (representing 5/7).
Beyond Comparison: Solving for Unknowns
Cross multiplication isn’t just for comparing fractions; it’s also a powerful tool for solving equations involving fractions. For example:
If x/5 = 9/15
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Cross Multiply:
- x * 15 = 15x
- 5 * 9 = 45
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Solve the Equation:
- 15x = 45
- x = 45 / 15
- x = 3
Conclusion: Why Cross Multiplication Works
Cross multiplication provides a shortcut for comparing fractions by leveraging the principles of equivalent fractions. By creating a common denominator and then comparing numerators, we can accurately determine which fraction is larger or if they are equal. This same principle extends to solving equations with fractions, making cross multiplication a versatile tool in mathematics.
