Exponents        Single Var. Eq.        Multi-Var. Eq.        Word Problems        Factoring    Fractions        Ratios        Number Lines        Coordinate Plane       Square Roots        Scientific Not. On this page, we hope to clear up problems that you might have with fractions and their uses in Algebra.  Fractions are continually being utilized in math, so they're quite an important subject to understand.  Click any of the links below to start understanding fractions better! Multiplication of fractions LCM, or Lowest Common Multiple Addition of fractions Complex fractions Quiz on Fractions In this section we will help you understand how to multiply fractions that contain rational expressions (expressions containing variables).  If you still need help with fractions that only contain numbers, click here. When fractions are multiplied, they are multiplied by multiplying the numerators by each other and the denominators by each other.  No cross-multiplication is involved!  Always remember that variables stand for numbers!  Because of that fact, nothing changes when dealing with variables in a multiplication problem with fractions that have rational expressions in them.  ``` _ _ 1. Expand: x^2| x^2 3y^3 | ---| --- - ---- | y^2|_ y m _| Solution: Two multiplications are supposed to be done. x^2 x^2 You have to first multiply --- by --- and then y^2 y x^2 -3y^3 you have to multiply --- by -----. y^2 m That gives you: (x^2)(x^2) (x^2)(3y^3) ---------- - -----------. (y^2)y (y^2)m Lastly, you simplify both expressions since answers are not considered correct in Algebra unless they are simplified. After simplification (multiply terms together and cancel things out if possible), you get the answer - x4 3yx2 -- - ---- y3 m``` When adding or subtracting fractions from other numbers, and especially other fractions, you must have the denominators of each fraction be the same expression.  Addition and subtraction are discussed further down on this page, but we will help you understand the Lowest Common Multiple in this section. Specifically, we will help you understand how to find LCMs of algebraic expressions. ``` 1. Find the LCM of 15a2b and 10ab3. Solution: Write the expressions as products of prime and literal factors. 15a2b 10ab3 3 * 5 * a * a * b 2 * 5 * a * b * b * b From the listing of prime and literal factors, take the groups of factors with the most instances of that factor. For example, there are three bs in 10ab3's factors, therefore, three bs are listed in the LCM. The LCM is the following: 2 * 3 * 5 * a * a * b * b * b = 30a2b3 ``` The addition or subtraction of fractions is complicated by the fact that the denominators must be the same before the fractions can be added or subtracted.  As we move toward more complex Algebra, you will come across fractions that have polynomials in them.  If the denominator is a polynomial, this polynomial must be a factor in the least common multiple. ```1. Add: a b - + ------- x (x + y) Solution: First, find the LCM of the denominators, which will become the new denominator. -------- + -------- x(x + y) x(x + y) So that the problem does not change, the numerator of each term has to be multiplied by the same quantity that its respective denominator was. The original denominator of the first term was x, and it has been multiplied by (x + y), so the original numerator, a, must be multiplied by (x + y), too. a(x + y) -------- + -------- x(x + y) x(x + y) The original denominator of the second term was (x + y), but it was multiplied by x, so the original numerator must also be multiplied by x. Now, the fractions can be added together. a(x + y) xb a(x + y) + xb -------- + -------- = ------------- x(x + y) x(x + y) x(x + y) This is one of the rare times in Algebra that there are multiple forms of the correct answer. In this case, you can multiply out the numerator and/or denominator if you want, and since doing that does not help you simplify the answer any further, they are also correct forms of the answer. ``` Another name for a fraction divided by a fraction is a complex fraction.  Complex fractions are typically shown as follows:  a  -  b ---  c  -  d where b, c, and d do not equal 0.  This is the same as (a/b)/(c/d).  To solve these fractions, you will need to multiply the numerator, or the first term of the problem by the reciprocal of the denominator, or the second term of the problem. ```1. Simplify: a - b --- c (b, c do not equal 0) Solution: Multiply the numerator (a/b) by the reciprocal of the denominator, which is (1/c). a 1 a - * - = -- b c bc ``` Take the Quiz on fractions.  (Very useful to review or to see if you've really got this topic down.)  Do it!

Math for Morons Like Us - Algebra: Fractions
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