Cross multiplication is a simple yet powerful method for comparing fractions. This technique allows you to quickly determine whether two fractions are equal or which one is larger. This method is particularly helpful when dealing with fractions that are difficult to compare visually or simplify.
Understanding Cross Multiplication
Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa. Let’s illustrate with two fractions: (a/b) and (c/d).
To cross multiply:
- Multiply a (numerator of the first fraction) by d (denominator of the second fraction). This gives you ad.
- Multiply b (denominator of the first fraction) by c (numerator of the second fraction). This gives you bc.
Comparing Fractions with Cross Multiplication
After performing the cross multiplication, compare the two resulting products (ad and bc):
- If ad = bc: The fractions are equal (a/b = c/d).
- If ad > bc: The first fraction (a/b) is greater than the second fraction (c/d).
- If ad < bc: The first fraction (a/b) is less than the second fraction (c/d).
Example 1: Comparing Two Fractions
Let’s compare the fractions 4/26 and 7/32.
- Cross multiply:
- 4 * 32 = 128
- 7 * 26 = 182
- Compare the products: 182 > 128.
Conclusion: Since 182 (representing 7/32) is greater than 128 (representing 4/26), we conclude that 7/32 is greater than 4/26.
Example 2: Solving for an Unknown Variable
Cross multiplication can also be used to solve for an unknown variable in an equation involving fractions. 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 provides a straightforward method to compare fractions and solve for unknown variables in fraction equations. By multiplying the numerator of one fraction by the denominator of the other, and comparing the resulting products, you can easily determine the relationship between two fractions. This technique is a valuable tool for anyone working with fractions.
