XV Operations on Algebraic Fractions

Same rules as arithmetic operations on fractions.

Addition and Subtraction

x12a2by18ab2\frac {x}{12a^2b} - \frac {y}{18ab^2}

  1. Find LCD
  2. For each denominator find the required factors to make them equal to the LCD
  3. Multiply the numerators by these factors
  4. Add or subtract the resulting numerators on top of the LCD

Step 1


Step 2

  1. Term: 3b3b
  2. Term: 2a2a

Step 3

  1. Term: x×3×bx \times 3 \times b
  2. Term: y×2×ay \times 2 \times a

Step 4

This yields the result of the above expression: 3bx2ya36a2b2\frac {3bx - 2ya }{36a^2b^2}

Alternatively multiply the numerators by the LCD, leaving the denominators alone, simplify the fractions and put the resulting terms on top of the LCD.

Multiplication and Division

In case of multiplication simply multiply both the terms of both fractions. In case of division multiply the reciprocal of one term with the other.

6ax414x2y2×2y33a4\frac {6ax^4}{14x^2y^2} \times \frac {2y^3}{3a^4} =12ax4y342a4x2y2= \frac {12ax^4y^3}{42a^4x^2y^2}
simplify (variable with negative exponent is moved to the other side of the fraction with the sign reversed)
12ax4y342a4x2y2=2x2y7a3\frac {12ax^4y^3}{42a^4x^2y^2} = \frac {2x^2y}{7a^3}

Taking the reciprocal of both sides

If two fractions are equal, their reciprocates are also equal. This can be used in solving equations involving fractions. Take for example

V=13πr2hV = \frac 1 3 \pi r^2 h and solve for hh

Dividing by hh yields Vh=13πr2\frac V h = \frac 1 3 \pi r^2 which is equal to Vh=πr23\frac V h = \frac {\pi r^2}{3}

Now VV needs to be moved to the denominator so that it can be moved to the other side by multiplication. This can be achieved by taking the reciprocal of both sides of the equation:

Vh=πr23\frac V h = \frac {\pi r^2}{3} is the same as hV=3πr2\frac h V = \frac {3}{\pi r^2}

From that equation multiply by VV to solve for hh:

h=3Vπr2h = \frac {3V}{\pi r^2}

Complex Problems

Distributing Results of Operations

12(ab2a+b3)12 ( \frac {a-b}{2} - \frac {a+b}{3})

  1. Subtract fractions within brackets
  2. Multiply the result with the coefficient
  3. Simplify (if the denominator is a common factor with both numerators)

Step 1

3a3b62a+2b6=3a3b6+2a2b6\frac {3a-3b}{6} - \frac {2a+2b}{6} = \frac {3a-3b}{6} + \frac {-2a - 2b}{6}
=a5b6= \frac {a-5b}{6}

Step 2

121×a5b6=12a60b6\frac {12}{1} \times \frac {a-5b}{6} = \frac {12a-60b}{6}

Step 3

2a10b1=2a10b\frac {2a-10b}{1} = 2a-10b

Distribution of exponents

(x)3=(x1)3=3x33(-x)^3 = (\frac{-x}{1})^3 = \frac {-3x^3}{3}


x3=x31x^3 = \frac {x^3}{1}


Expressions may be factored to isolate subjects:

P+12bP=Q(b+12)P+\frac 1 2 bP = Q(b+\frac 1 2 )
P(1+12b)=Q(b+12)P(1+\frac 1 2 b) = Q(b+\frac 1 2)

Moving factors outside of fractions

If the numerator is a single term expression, any of its factors may be pushed out to become factors of the whole fraction:

abcd=b(acd)\frac{ab}{cd} = b(\frac{a}{cd})

P(1+12b)=Q(b+12)P(1+\frac 1 2 b) = Q(b + \frac 1 2)
P=Q(b+12)1+12bP = \frac {Q(b+\frac 1 2)}{1+\frac 1 2 b}
P=b+121+12bQP = \frac {b+\frac1 2}{1+\frac 1 2 b}Q


Only factors may be cancelled, never whole terms

x+6x+321\frac {x+6}{x+3} \ne \frac 2 1

A factor may only cancel out with one other factor

a(a+3b)(a+3b)b(a+2b)(a+3b)=a(a+3b)b(a+2b)\frac{a(a+3b)(a+3b)}{b(a+2b)(a+3b)} = \frac{a(a+3b)}{b(a+2b)}

Complex Fractions

abc=ab×1c=abc\frac{\frac a b}{c} = \frac a b \times \frac 1 c = \frac {a}{bc}