Cross multiplication is a handy trick for comparing fractions. But why does it work? This article explains the mathematical reasoning behind cross multiplication and demonstrates how to use it effectively.
Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other, and vice versa. The results of these multiplications allow you to determine if the fractions are equal or if one is larger. This method stems from the fundamental principle of equivalent fractions.
To understand the underlying principle, let’s consider two fractions: a/b and c/d. If these fractions are equal (a/b = c/d), then multiplying both sides of the equation by the same value maintains the equality. So, if we multiply both sides by the product of the denominators (b*d), we get:
(a/b) (bd) = (c/d) (bd)
Simplifying this equation leads to:
ad = cb
This is the core of cross multiplication. When you cross multiply, you’re essentially finding a common denominator and comparing the numerators. If ad equals cb, the original fractions are equivalent.
Comparing Fractions with Cross Multiplication
Example 1: Determining Which Fraction is Greater
Let’s compare 4/26 and 7/32.
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Cross multiply:
- 4 * 32 = 128
- 7 * 26 = 182
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Compare the products: Since 182 > 128, we know that 7/32 > 4/26.
Remember, the product represents the corresponding fraction. 128 represents 4/26, and 182 represents 7/32.
Solving for Unknowns with Cross Multiplication
Cross multiplication isn’t just for comparing fractions; it can also solve for unknown variables in equations involving fractions.
Example 2: Solving for x
Consider the equation 9/16 = x/27.
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Cross multiply:
- 9 * 27 = 243
- 16 * x = 16x
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Set up the equation: 243 = 16x
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Solve for x: Divide both sides by 16 to isolate x. x = 243/16
Conclusion
Cross multiplication offers a simple yet powerful way to compare fractions and solve equations involving fractions. It works because it’s based on the principle of creating equivalent fractions with a common denominator. By comparing the resulting numerators, you can accurately determine the relationship between the original fractions. This technique is a valuable tool for anyone working with fractions.
