## Fractions. Variables in the Denominator

### Variables in the Denominator

Fraction problems get more difficult to solve when there is a variable in the denominator. To solve, find the least common denominator (LCD) of the fractions, and multiply both sides of the equation by it.
E X A M P L E :
Solve $\large \; 4-\frac{1}{x}=\frac{6}{2x}.$
The LCD of $\frac{1}{x}$ and $\frac{6}{2x}$ is $2x$.
$2x\left(4-\frac{1}{x} \right)=2x\left(\frac{6}{2x} \right).$
$8x-2=6.$
$8x=8.$
$x=1.\; \; \;$ (Answer)
Sometimes multiplying both sides of an equation by the LCD transforms the equation into an equation that is NOT equivalent to the original one. Multiplying both sides of an equation by a polynomial may introduce extraneous roots that do not satisfy the original equation. It is crucial to go back and check your answer in the original fractional equation.
E X A M P L E :
Solve $\large \frac{2}{\left(x^{2}-7x+10 \right)}=\frac{x-1}{x-5}.$
Factor $x^{2}-7x+10$ into $(x-2)(x-5)$. Now multiply both sides of the equation by the LCD of the fractions, $(x-2)(x-5)$.
$\frac{2}{\left(x^{2}-7x+10 \right)}\times (x-2)(x-5)=\frac{x-1}{x-5}\times (x-2)(x-5).$
$2= (x-2)(x-5).$
$2= x^{2}-3x+2.$
$0= x^{2}-3x.$
$0= x\left(x-3 \right).$
$x=0$ or $x=3$. (Answer)
Because you multiplied both sides of the equation by a polynomial $(x-2)(x-5)$, check to ensure the equation does not have extraneous roots. Substituting x = 0 into the original equation results in $\frac{2}{10}=\frac{-1}{-5}$ or 1/5=1/5. Substituting x = 3 into the original equation results in 2/−2 = 2/−2 or −1 = −1. Both answers check, so they are not extraneous.
Proportions are another type of problem that may have variables in the denominator. A proportion is an equation that sets two ratios (fractions) equal to each other. Don’t worry about finding least common denominators when solving a proportion, simply cross -multiply.
E X A M P L E :
$\large \frac{10}{x+4}=\frac{6}{x}.$
$10x=6\left(x+4 \right).$
$10x=6x+24.$
$4x=24.$
$x=6.$ (Answer)

## Fractions. Using Mixed Numbers and Improper Fractions

### Using Mixed Numbers and Improper Fractions

A mixed number represents the sum of an integer and a fraction.
$3\frac{1}{4}=3+\frac{1}{4}.$
In fractional form: $3\frac{1}{4}=\frac{4\times 3+1}{4}=\frac{13}{4}.$
When $\; 3\frac{1}{4}$ is written as the fraction $\; 3\frac{13}{4}$ it is called an improper fraction.
Improper fractions are fractions whose numerator is greater than the denominator.
It is often easier to change mixed numerals to improper fractions when
simplifying an expression.

## Fractions. Multiplying Fractions

### Multiplying Fractions

To multiply fractions, simply multiply straight across.

$\displaystyle \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}.$

E X A M P L E :
Simplify $\displaystyle \; \frac{2}{5}\times \frac{3}{6}\times \frac{4}{7}.\;$
$\displaystyle \;\;\; \; \;\; \; \; =\frac{2\times 3\times 4}{5\times 6\times 7}.\;$
$\displaystyle \;\;\; \; \;\; \; \; =\frac{24}{210}.\;$ Divide by a common factor to simplify.
$\displaystyle \;\;\; \; \;\; \; \; =\frac{4}{35}.\;$ (Answer)
You can also simplify the fractions before multiplying to save time.

## Fractions. Least Common Denominator

### Least Common Denominator

The least common denominator (LCD) of two or more fractions is the least
common multiple (LCM) of their denominators. To find the LCD:
1. Factor each denominator completely and write as the product of prime factors.
(Factor trees are usually used for this.)
2. Take the greatest power of each prime factor.
3. Find the product of these factors.

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## Fractions. Simplifying Fractions

### Simplifying Fractions

Fractions are in simplest form when the numerator and denominator have no common factor other than 1. To simplify a fraction, factor both the numerator and denominator.
Don't cancel terms that are not common factors. Below is a common mistake: