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Fractions

There's a lot of stuff in this topic! Read through each of the subtopics below carefully!

We use the same rules we have for arithmetic fractions when we are working with algebraic ones:

Rules for Fractions
1. (a/c)+(b/c)=(a+b)/c add numerators when denominators are equal, or
2. (a/b) + (c/d) = a(d)/(bd) + (b)c/(bd) = (ad+bc)/bd find a common denominator
3. (a/b)(c/d) = (ac)/(bd) multiply numerators and denominators for a product
4. (a/b)/(c/d) = (ad)/(bc) to divide a fraction, multiply by its reciprocal
  (a/b)/c = (a/b)(1/c) = a/(bc) this is the same as rule 4: we're multiplying by 1/c
  a/(b/c) = (a/1)/(b/c) = (a/1)(c/b) = ac/b also the same as rule 4.
(We assume that no denominators are zero.)

In no case can we divide by zero: (a/0) is undefined, and (0/0) is also undefined. (a/a) = 1. And the sign of a fraction is changed by changing the sign of either the numerator or the denominator, but not both:
-(a/b) = -a/b = a/(-b)

Examples
For each, we perform the indicated operations and express the answers as a single fraction. Carefully note which rules for fractions are used at each step.
  1. 4/(x^2+1)-(1-x)/(x^2+1) = (4-(1-x))/(x^2+1) = (3+x)/(x^2+1)
  2. M/(M^2-2M-3) + 1/(M^2-2M-3) = (M+1)/(M^2-2M-3) = (M+1)/((M+1)(M-3)) = 1/(M-3)
  3. (-H^2P/17)((PH^(1/3))^2/K^(-1)) = (-H^2P(P^2 H^(2/3)))/(17K^(-1)) = -H^(8/3) P^3 K/17
  4. (2z/w)/(w(w-3z))=(2z/w)(1/(w(w-3z)))=2z/(w^2(w-3z))
  5. 2x^(-1/2)+sqrt(x)/3 = 2/sqrt(x) + sqrt(x)/3 = (2*3 + sqrt(x)*sqrt(x))/3sqrt(x) = (6+x)/(3sqrt(x)) = (6+x)/(3x^(1/2))

Finding a common denominator and canceling

As in the last of the examples above, we can multiply (or divide) both the numerator and denominator of a fraction by the same non-zero factor without changing the fraction's value. This is equivalent to multiplying by 1. We are using this rule when we add or subtract fractions with different denominators. For example, to add x/(3a) and 1/a we multiply (1/a)(3/3) = 3/(3a). Then
x/(3a)+1/a = x/(3a)+3/(3a) = (x+3)/3a

Be careful with the following
We can multiply (or divide) the numerator and denominator of a fraction by the same thing because this is the same as multiplying by 1, which doesn't change the value of the expression. However, we cannot perform any operation which changes the value of the expression. For example, we cannot add the same value to the numerator or denominator of a fraction, square both, etc., without changing the fraction.

The reverse of multiplying the numerator and denominator of a fraction by something is canceling common factors that appear in both numerator and denominator:
ac/(bc) = (a/b)(c/c) = (a/b)(1) = (a/b).

Examples
Again, note carefully which rules for fractions we are using at each step, and what common denominators we use or factors we cancel.
  1. M/(M^2-M+3) + (1/M) = M(M)/(M(M^2-M+3)) + (M^2-M+3)/(M(M^2-M+3)) = (M^2 + M^2 - M + 3)/(M(M^2-M+3)) = (2M^2-M+3)/(M(M^2-M+3))
  2. 3 - (1/(x-1)) = 3((x-1)/(x-1)) - 1/(x-1) = (3(x-1) - 1)/(x-1) = (3x - 3 - 1)/(x-1) = (3x - 4)/(x-1)
  3. 2/(x^2+x) + x/(x+1) = 2/(x(x+1)) + x/(x+1) = 2/(x(x+1)) + x*x/(x(x+1)) = (2+x^2)/(x(x+1))
  4. 2x/(4y) = (1/2)(x/(2y)) = x/(2y)
  5. (2+x)/(2+y) cannot be reduced further
  6. (5n-5)/(1-n) = 5(n-1)/((-1)(n-1)) = -5
  7. (x^2(4-2x)-(4x-x^2)(2x))/x^4 = (x^2(4-2x)-(4-x)(2x^2))/x^4 = ((4-2x)-2(4-x))/x^2 x^2/x^2 = (4-2x-8+2x)/x^2 = -4/x^2 = -(4/x^2)

Complex fractions

A complex fraction is one whose numerator or denominator (or both) contains one or more fractions. To simplify a complex fraction, we change the numerator and enominator to single fractions and then divide.

Examples
  1. ((1/(x+h))-(1/x))/h =(x-(x+h))/(x(x+h))/h = (-h/(x(x+h)))/h = (-h/(x(x+h)))(1/h) = -1/(x(x+h))
  2. (a+b)/(a^(-2)-b^(-2)) = (a+b)/(1/a^2 - 1/b^2) = (a+b)/(b^2-a^2)/(a^2b^2) = ((a+b)/1)(a^2b^2/(b^2 - a^2)) = (a+b)(a^2b^2)/((b+a)(b-a)) = (a^2b^2)/(b - a)

Splitting expressions

We can reverse the rule for adding fractions to split up an expression into two fractions:
(a+b)/c = (a/c) + (b/c)
 Sometimes we can alter the form of the fraction even further by creating a duplicate of the denominator in the numerator, as shown in the second example below.

Examples
Be sure that you can see what rules we're using at each step of the examples. In particular, notice how we modify the second example and then use the rules.
  1. (3x^2+2)/x^3 = (3x^2/x^3)+2/x^3 = 3/x+2/x^3
  2. (x+3)/(x-1) = ((x-1+1)+3)/(x-1) = ((x-1)+4)/(x-1) = (x-1)/(x-1)+4/(x-1) = 1 + 4/(x-1)
Practice

Be sure you've gone through each step in the examples above before doing these. Then work the practice problems. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Combine the given fractions into the form
    (a x + b)/(c x + d)
    (Enter 0 for a or c if there is no x term in the answer, and similarly for b and d).
    a x   +   b   +   g x   +   h   =   x   +  
    . . .
    c x   +   d c x   +   d x   +  
    (Enter answers, then click: . Answer message: )
  2. Combine the given fractions into the form
    (a x + b)/(c x^2 + d x + g)
    (Enter 0 for any value that does not exist in the answer.)
    j   -   k   =     x   +  
    . . .
    c x   +   d m x   +   n x ^2   +   x   +  
    (Enter answers, then click: . Answer message: )
  3. Simplify the given complex fraction into the form
    a/b
    (reduce the fraction you get to lowest terms!)
    o   -   p   =  
    .
    .
    q   +   r
    . .
    s s
    (Enter answers, then click: . Answer message: )
  4. Rewrite in the form
    1 + (ax+b)/(cx+d)
    u x   +   v   =   1   +  
    . .
    u x   +   w x   +  
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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precal: 4.1 - fractions
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