## Tuesday, March 3, 2009

### PREP FOR GMAT: FRACTIONS

A fraction is a number that represents a ratio or division of two numbers a and b with b not equal to 0.
It is typically written in the form a/b.

a <<<<< numerator.
--
b<<<<< denominator.

For example, 3/4 is a fraction whose numerator is 3 and denominator is 4.

The denominator indicates the total number of equal parts that something is divided into, whereas the numerator indicates the number of these parts that are taken.
For example, 2/3 indicates taking 2 out of a total of 3 equal parts.

Types of fractions:

Proper Fractions :A proper fraction is a fraction with a numerator smaller than the denominator, and fraction value less than 1.For example, 2/5
Improper Fractions :An improper fraction is a fraction with a numerator greater than or equal to the denominator, and therefore fraction value greater than or equal to 1.For example, 5/2
Mixed Numbers :A mixed number comprises a natural number and a properfraction, and therefore always has a value greater than 1.For example, 5¾ (read as 'five and three-fourths')

Equivalent Fraction;

Fractions represent a part of a whole.
To determine if two fractions are equivalent, multiply the denominator and numerator of one fraction so that the denominators of the two fractions are equal(preferably LCM) . As long as you multiply or divide both the numerator and denominator of a fraction by the same nonzero number, you will not change the overall value of the fraction.

Reducing Fractions

Reducing fractions makes calculations simpler. Anyting of huge size would be scary to us but we will take advantage on smaller things. To reduce a fraction to its lowest terms, divide the numerator and denominator by their GCF. For example450/600, for , the GCF of 450 and 600 is 150. So the fraction reduces down to 3/4,
A fraction is in its simplest, totally reduced form when the GCF of its numerator and denominator is 1. ie ,There is no other number but 1, that can divide into both 3 and 4, so is a fraction in its lowest form, reduced as far as it can go.
Note we can use Canceling (striking or crossing out) common factors which does not change the value of the fraction but gives equivalent fractions in reduced form.

When fractions have the same (like) denominator, add or subtract the numerators and place the result over the common denominator. Then simplify the new fraction if required.

To add fractions with the same denominators, all you have to do is add up the numerators and keep the denominator the same:
Subtraction works similarly. If the denominators of the fractions are equal, just subtract one numerator from the other and keep the denominator the same:
Fractions can be negative too:

When fractions do not have the same denominator, addition and subtraction are performed after each fraction is expressed as an equivalent fraction with the (least) common denominator given by the LCM of the denominators. Using the least common multiple (LCM) typically reduces the computational effort.

5/3 +7/4

You could try to find the common denominator by multiplying the denominators(3*4). The key is to multiply diagonally and up, which in this case means from the 5 to the 4 and also from the 7 to the 3:

We add the products to get our numerator: 20 + 21 = 41. For the denominator, we simply multiply the two denominators to get:

Subtracting. Same basic deal, except this time we subtract the products that we get when we multiply diagonally and up.

Multiplying Fractions

The product of two fractions is merely the product of their numerators in numerator and the product of their denominators in denominator.

Dividing Fractions

Multiplication and division are inverse operations. It makes sense, then, that to perform division with fractions, all you have to do is flip the second fraction and then multiply.

Complex Fractions.

If the numerator and denominator of a fraction are themselves fractions, then it is called a complex fraction.This is similar to division , the rule is to invert and multiply. Take whichever fraction appears on the bottom of the complex fraction, or whichever fraction appears second if they’re written in a single line, and flip it over. Then multiply by the other fraction.

Mixed Numbers

A mixed number is an integer followed by a fraction . You have to know how to convert them into standard fraction form.
The method is to Multiply the integer of the mixed number by the denominator of the fraction part, and add that product to the numerator. This will be the numerator. Now, put that over the original denominator,to get the converted fraction.
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