Gre math Review 1 Arithmetic

1.2 Fractions

A fraction is a number of the form   a over b, where a and b are integers and

 b is not equal to 0. The integer a is called the numerator of the fraction, and b is called the denominator. For example,   negative 7 over 5 is a fraction in which   negative 7 is the numerator and 5 is the denominator. Such numbers are also called rational numbers.

If both the numerator a and denominator b are multiplied by the same nonzero integer, the resulting fraction will be equivalent to   a over b. For example,

 
the fraction negative 7 over 5 = the fraction with numerator negative 7 times 4 and denominator 5 times 4, which is equal to the fraction negative 28 over 20, and the fraction negative 7 over 5 is also equal to the fraction with numerator negative 7 times negative 1 and denominator 5 times negative 1, which is equal to the fraction 7 over negative 5

A fraction with a negative sign in either the numerator or denominator can be written with the negative sign in front of the fraction; for example,

 the fraction negative 7 over 5 = the fraction 7 over negative 5, which is equal to the negative of the fraction 7 over 5.

If both the numerator and denominator have a common factor, then the numerator and denominator can be factored and reduced to an equivalent fraction. For example,

 the fraction 40 over 72 = the fraction with numerator 8 times 5 and denominator 8 times 9, which is equal to the fraction 5 over 9.

To add two fractions with the same denominator, you add the numerators and keep the same denominator. For example,

  the negative of the fraction 8 over 5 + the fraction 5 over 11 = the fraction with numerator negative 8 + 5, and denominator 11, which is equal to the fraction negative 3 over 11, which is equal to the negative of the fraction 3 over 11.

To add two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator. For example, to add the two fractions  1 third and negative 2 fifths, use the common denominator 15:

 
1 third + negative 2 fifths = 1 third times 5 over 5, +, negative 2 fifths times 3 over 3, which is equal to 5 over 15 + negative 6 over 15, which is equal to the fraction with numerator 5 + negative 6, and denominator 15, which is equal to the negative of the fraction 1 over 15.

The same method applies to subtraction of fractions.

To multiply two fractions, multiply the two numerators and multiply the two denominators. Here are two examples.

Example A:  

The fraction 10 over 7 times the fraction negative 1 over 3 = the fraction with numerator 10 times negative 1 and denominator 7 times 3, which is equal to the fraction negative 10 over 21, which is equal to the negative of the fraction 10 over 21
Example B:  
The fraction 8 over 3 times the fraction 7 over 3 = the fraction 56 over 9

To divide one fraction by another, first invert the second fraction, that is, find its reciprocal; then multiply the first fraction by the inverted fraction. Here are two examples.

Example A:  
The fraction 17 over 8, divided by the fraction 3 over 4 = the fraction 17 over 8, times the fraction 4 over 3, which is equal to the fraction 4 over 8, times the fraction 17 over 3, which is equal to the fraction 1 over 2 times the fraction 17 over 3, which is equal to the fraction17 over 6
Example B:  
The fraction with numerator equal to the fraction 3 over 10 and denominator equal to the fraction 7 over 13 = the fraction 3 over 10, times the fraction 13 over 7, which is equal to the fraction 39 over 70

An expression such as   4 and 3 eighths is called a mixed number. It consists of an integer part and a fraction part; the mixed number   4 and 3 eighths means

 the integer 4 + the fraction 3 eighths. To convert a mixed number to an ordinary fraction, convert the integer part to an equivalent fraction and add it to the fraction part. For example,

 

the mixed number 4 and 3 eights = the integer 4 + the fraction 3 eighths, which is equal to the fraction 4 over 1, times the fraction 8 over 8, +, the fraction 3 over 8, which is equal to the fraction 32 over 8 + the fraction 3 over 8, which is equal to the ordinary fraction 35 over 8.

Note that numbers of the form   a over b, where either a or b is not an integer and   b is not equal to 0, are fractional expressions that can be manipulated just like fractions. For example, the numbers   pi over 2 and pi over 3 can be added together as follows.

 
pi over 2 + pi over 3 = pi over 2, times 3 over 3, +, pi over 3 times 2 over 2, which is equal to 3pi over 6, +, 2pi over 6, which is equal to 5pi over 6

And the number

 
the fraction with numerator equal to the fraction 1 over the positive square root of 2 and denominator equal to the fraction 3 over the positive square root of 5
can be simplified as follows.
 
the fraction with numerator equal to the fraction 1 over the positive square root of 2 and denominator equal to the fraction 3 over the positive square root of 5 = the fraction 1 over the positive square root of 2, times the fraction with numerator equal to the positive square root of 5 and denominator 3, which is equal to the fraction with numerator equal to the positive square root of 5 and denominator equal to 3 times the positive square root of 2