How to Multiply and Divide Algebraic Fractions

In arithmetic, multiplying and diving fractions is relatively straightforward. To multiply fractions, simply multiply the numerators, then multiply the denominators. Reduce to lowest terms if necessary. Dividing fractions requires an extra step. To divide fractions, first reverse the numerator and denominator of the second fraction. Then multiply as normal. An algebraic fraction has at least one variable in the denominator. Multiplying and dividing algebraic expressions follows a similar procedure to regular fractions, though you might need a few extra steps.

Difficulty: Moderate

Instructions

Multiplying Algebraic Fractions

1. Take an example problem: 3x/2y^2 times 6y/5x.

2. Multiply the numerators: 3x times 6y is 18xy.

3. Multiply the denominators: 2y^2 times 5x is 10xy^2. The new fraction is 18xy/10xy^2.

4. Look for terms you can cancel out. You can divide 18 and 10 by 2: 18/2 is 9 and 10/2 is 5. So you have 9xy/5xy^2.

5. Cancel out the x's and y's. x cancels out because it appears in both the numerator and denominator. y cancels out once because you have y in the numerator and y^2 (or yy) in the denominator. So you have 9/5y. This is the final answer.

6. Some of the more complicated problems may require you to factor out algebraic expressions and to simplify before solving. Take an example problem: (x^2 - y^2)/x^2 + 2xy + y^2 times (xy + y^2)/(x^2 - y^2)

7. Before multiplying the numerators and denominators, look for ways to simplify the expressions. You will see that x^2 - y^2 appears in the numerator of the first term and in the denominator of the second term. Go ahead and cancel out x^2 - y^2.

8. Canceling out x^2 - y^2, rewrite the problem as 1/(x^2 + 2xy + y^2) times xy + y^2/1.

9. Take the expression xy + y^2. Factor out the y to get y(x + y).

10. Rewrite the problem as 1/(x^2 + 2xy + y^2) times y(x + y)/1.

11. Factor out x^2 + 2xy + y^2 to get (x + y)(x + y) or (x + y)^2. You can check this by multiplying out the parentheses with the FOIL method. Multiply the first, outer, inner, and last terms: (x + y)(x + y) = x^2 + xy + xy + y^2 = x^2 + 2xy + y^2.

12. Rewrite the problem as 1/(x + y)^2 times y(x + y)/1.

13. Cancel out an (x + y) in both terms. You can do this because (x + y) appears in the denominator of the first term and in the numerator of the second term.

14. Rewrite the problem as 1/(x + y) times y/1.

15. Multiply the numerators: 1 times y is y.

16. Multiply the denominators: (x + y) times 1 is (x + y).

17. Place the new numerator y over the new denominator (x + y). The final answer is y/(x + y).

Divinding Algebraic Fractions

18. Take another example: 3x/8 divided by 9x/16.

19. Invert the second fraction: 9x/16 becomes 16/9x.

20. Rewrite the problem as 3x/8 times 16/9x.

21. Multiply the numerators: 3x times 16 is 48x.

22. Multiply the denominators: 8 times 9x is 72x. The new fraction is 48x/72x.

23. Reduce the numbers to lowest terms. You will see that 48 and 72 are both divisible by 24. 48/24 is 2 and 72/24 is 3. So you have 2x/3x.

24. Cancel out the x's. x cancels out because it appears once in the numerator and the denominator. 2x/3x cancels out to 2/3. This is the final answer.

Factoring in Algebraic Fractions

25. Take an example problem: (x^2 - y^2)/x^2 + 2xy + y^2 times (xy + y^2)/(x^2 - y^2). Some of the more complicated problems may require you to factor out algebraic expressions and to simplify before solving.

26. Before multiplying the numerators and denominators, look for ways to simplify the expressions. You will see that x^2 - y^2 appears in the numerator of the first term and in the denominator of the second term. Go ahead and cancel out x^2 - y^2.

27. Canceling out x^2 - y^2, rewrite the problem as 1/(x^2 + 2xy + y^2) times xy + y^2/1.

28. Take the expression xy + y^2. Factor out the y to get y(x + y).

29. Rewrite the problem as 1/(x^2 + 2xy + y^2) times y(x + y)/1.

30. Factor out x^2 + 2xy + y^2 to get (x + y)(x + y) or (x + y)^2. You can check this by multiplying out the parentheses with the FOIL method. Multiply the first, outer, inner, and last terms: (x + y)(x + y) = x^2 + xy + xy + y^2 = x^2 + 2xy + y^2.

31. Rewrite the problem as 1/(x + y)^2 times y(x + y)/1.

32. Cancel out an (x + y) in both terms. You can do this because (x + y) appears in the denominator of the first term and in the numerator of the second term.

33. Rewrite the problem as 1/(x + y) times y/1.

34. Multiply the numerators: 1 times y is y.

35. Multiply the denominators: (x + y) times 1 is (x + y).

36. Place the new numerator y over the new denominator (x + y). The final answer is y/(x + y).

Resources

• Themathpage.com: Multiplying Algebraic Fractions

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