Cross multiplication is a common method used to compare fractions. It involves multiplying the numerator of one fraction by the denominator of the other, and vice versa. This process helps determine if two fractions are equal or if one is greater than the other. This is especially useful when dealing with fractions that are difficult to compare visually or by finding a common denominator.
How Does Cross Multiplication Work for Fraction Comparison?
Let’s say we have two fractions: a/b and c/d. To cross multiply:
- Multiply the numerator of the first fraction (a) by the denominator of the second fraction (d). This gives you ad.
- Multiply the numerator of the second fraction (c) by the denominator of the first fraction (b). This gives you cb.
Now, compare the two products (ad and cb):
- If ad = cb: The fractions are equivalent (a/b = c/d).
- If ad > cb: The first fraction (a/b) is greater than the second fraction (c/d).
- If ad < cb: The first fraction (a/b) is less than the second fraction (c/d).
Examples of Comparing Fractions with Cross Multiplication
Let’s illustrate with a couple of examples:
Example 1: Compare 4/26 and 7/32.
- Cross multiply: 4 32 = 128 and 7 26 = 182.
- Compare the products: 128 < 182.
Therefore, 4/26 is less than 7/32.
Example 2: Compare 3/5 and 6/10.
- Cross multiply: 3 10 = 30 and 6 5 = 30.
- Compare the products: 30 = 30.
Therefore, 3/5 is equal to 6/10.
Using Cross Multiplication to Solve for Unknowns
Beyond comparing fractions, cross multiplication is also valuable for solving equations involving fractions with an unknown variable.
Example: Solve for x in the equation 9/16 = x/27.
- Cross multiply: 9 27 = 243 and 16 x = 16x.
- This gives you the equation: 243 = 16x.
- Solve for x by dividing both sides by 16: x = 243/16.
Conclusion
Cross multiplication offers a straightforward method for comparing fractions and solving for unknown variables in fractional equations. By understanding the principles of cross multiplication, you can confidently tackle fraction-related problems in mathematics. It provides a clear and efficient way to determine the relationship between two fractions or to find the value of an unknown variable.
