Cross multiplication in fractions is a method primarily used for comparing fractions or solving proportions, not for performing operations like addition or subtraction directly. However, one of the references does mention using it as the first step when adding fractions, which we will address below.
Let's explore how cross multiplication works:
Understanding Cross Multiplication
Cross multiplication involves multiplying the numerator of one fraction by the denominator of another fraction. This produces two new values that you can compare.
Steps for Cross Multiplication
- Set up the fractions: You need two fractions, typically written as a/b and c/d.
- Multiply diagonally: Multiply the numerator of the first fraction (a) by the denominator of the second fraction (d). This gives you 'ad'.
- Multiply the other way diagonally: Multiply the denominator of the first fraction (b) by the numerator of the second fraction (c). This gives you 'bc'.
- Compare results: You now have two values, 'ad' and 'bc'. Compare these two values to determine the relationship between the original fractions.
Applications of Cross Multiplication
Comparing Fractions
- If 'ad' is greater than 'bc', then the fraction a/b is greater than the fraction c/d.
- If 'ad' is less than 'bc', then the fraction a/b is less than the fraction c/d.
- If 'ad' is equal to 'bc', then the fractions a/b and c/d are equal.
Example:
Let's compare 2/3 and 3/5:
- Multiply 2 by 5 (2 * 5 = 10)
- Multiply 3 by 3 (3 * 3 = 9)
- Since 10 > 9, then 2/3 > 3/5
Solving Proportions
Cross multiplication is also commonly used to solve proportions (two equal ratios):
If a/b = c/d, then ad = bc. If you know three of the values (a, b, and c, for example) you can solve for the fourth (d).
Cross Multiplication When Adding Fractions
The reference states that you can use cross multiplication as a first step to add fractions. The reference describes multiplying diagonally to get a new numerator, then multiplying the two bottom numbers to get a new denominator. This approach is not for cross-multiplication, but rather to find the numerator and denominator after adding two fractions. Here is how this works:
- Cross Multiply for Numerator: For example, with 1/3 + 2/5, you would multiply 15=5 and 32=6.
- Add Cross Multiplied Results: Then, add 5 + 6 = 11. This is the new numerator.
- Multiply Denominators: Multiply 3 * 5 = 15, this is the new denominator.
- Result: The sum of these two fractions is 11/15.
Summary
Cross multiplication is a useful technique for comparing fractions and solving proportions by producing two new values to compare, not for directly adding, subtracting, multiplying or dividing fractions. Keep in mind that while the reference touches on its usage as a preliminary step when adding fractions, the reference is actually using cross multiplication to create a numerator and denominator to add fractions and does not use it to solve by multiplying.
