# Fractions

A fraction consists of two parts: a numerator and a denominator.*proper*and is less than one.

For example: 1/2, 4/5, and 3/Ï€ are all proper fractions and therefore less than 1.

If the numerator is larger than the denominator, the fraction is called

*improper*and is greater than 1.

For example: 3/2, 5/4, and Ï€/3 are all improper fractions and therefore greater than 1.

An improper fraction can be converted into a

*mixed fraction*by dividing its denominator into its numerator.

For example, since 2 divides into 7 three times with a remainder of 1, we get

To convert a mixed fraction into an improper fraction, multiply the denominator and the integer and then add the numerator. Then, write the result over the denominator.

For example, .

**Strategy:**To compare two fractions, cross-multiply. The larger number will be on the same side as the larger fraction.

9/10 | 10/11 |

Cross-multiplying gives 9 ** ^{.}** 11 versus 10

**10, which reduces to 99 versus 100. Now, 100 is greater than 99. Hence, 10/11 is greater than 9/10.**

^{.}

**Strategy:**Always reduce a fraction to its lowest terms.

*If x â‰ â€“1, then =*

*0**1**2**4**6*

**Strategy:** To solve a fractional equation, multiply both sides by the LCD (lowest common denominator) to clear fractions.

*If , what is the value of x in terms of y?*

*3 â€“ y**3/y**3y*^{2}

*x*â€“3:

Cancel the (x â€“ 3)'s on the left side of the equation: | x + 3 = (x â€“ 3)y |

Distribute the y: | x + 3 = xy â€“ 3y |

Subtract xy and 3 from both sides: | x â€“ xy = â€“3y â€“ 3 |

Factor out the x on the left side of the equation: | x(1 â€“ y) = â€“3y â€“ 3 |

Finally, divide both sides of the equation by 1 â€“ y: | |

Hence, the answer is (D). |

**Note: **Complex Fractions: When dividing a fraction by a whole number (or vice versa), you must keep track of the main division bar:

** **But** **

- 6
- 3
- 1/3
- 1/6
- 1/8

The answer is (D).

*If z â‰ 0 and yz â‰ 1, then*

The answer is (D).

**Note: **Multiplying fractions is routine: merely multiply the numerators and multiply the denominators: .

**For example,**

**.**

**Note: **Two fractions can be added quickly by cross-multiplying:

â€‹

*â€“5/4**â€“2/3**â€“1/4**1/2**2/3*

*Which one of the following equals the average of x and *

*?*

The average of *x* and 1/*x* is .

Thus, the answer is (B).

**Note:**To add three or more fractions with different denominators, you need to form a common denominator of all the fractions.

For example, to add the fractions in the expression , we have to change the denominator of each fraction into the common denominator 36 (note, 36 is a common denominator because 3, 4, and 18 all divide into it evenly).

**Note:**To find a common denominator of a set of fractions, simply add the largest denominator to itself until all the other denominators divide into it evenly.

**Note:**Fractions often behave in unusual ways: Squaring a fraction makes it smaller, and taking the square root of a fraction makes it larger.

**Caution:**This is true only for proper fractions, that is, fractions between 0 and 1.)

and 1/9 is less than 1/3. Also and 1/2 is greater than 1/4.

**Note:**You can cancel only over multiplication, not over addition or subtraction.