Fractions

From wikinotes

Components

Fractions

 numerator
-------------
 denominator

Ratios/Rates

Ratios are just another way of expressing fractions.
ex: 34:3 or 34 newspapers / 3 hrs

Reciprocals

The reciprocal of is .

Operations

Addition/Subtraction

  • find the lowest common denominator (see above) and top numbers
  • in each fraction, multiply the top/bottom by the same multiple it takes for the bottom make the bottom equal to 12.
  • add/subtract the numbers at the top of the faction

illustrative expanded version


Multiplication

Whole Numbers

Multiplication of whole numbers affects the top number.
It expresses sum of several fractions.

+-+-+-+-+-+
|x|x| | | |  # 2/5
+-+-+-+-+-+

# 2 * 2/5  == 4/5
+-+-+-+-+-+   +-+-+-+-+-+     +-+-+-+-+-+
|x|x| | | | + |x|x| | | |  =  |x|x|x|x| |
+-+-+-+-+-+   +-+-+-+-+-+     +-+-+-+-+-+

Fractions

Multiplication between fractions represents a fraction of a fraction.
Numbers on the top are multiplied against each other,
and numbers on the bottom are also multiplied against each other.

+-+-+-+-+-+
|x|x| | | |  # 2/5
+-+-+-+-+-+

+-+-+-+-+-+
|x|x| | | |
+-+-+-+-+-+  # 2/5 * 1/2  == 2/10
| | | | | |
+-+-+-+-+-+

Mixed Numbers

Division

Whole Numbers

Fractions

Methods

Finding the Lowest Common Denominator (LCM)

Lowest shared Multiple of Larger Number

  • find the lowest number divisible by both 4 and 6
    by listing the multiples of the larger number, and seeing if it is a multiple of the smaller number.
  • in this case is the smallest common denominator

Prime Factorization

  • Gradually decompose the number by dividing it by prime numbers
    (starting with smallest, gradually increasing in value)
  • Repeat until the number is expressed as several multiplications of prime numbers
 16
 | \
 2  8        # smallest prime number 16 is divisible by
    | \
    2  4     # smallest prime number 8 is divisible by
       | \
       2  2  # smallest prime number 4 is divisible by

You can use the prime factorization of 2x numbers to find the lowest common denominator.

8            50
| \          | \
2  4         5  10
   | \          | \
   2  2         2  5

If you multiply the prime numbers,
removing numbers that are shared in both,
you'll get the lowest common denominator.

# note that below, 1x '2' has been removed, since it is used in both
 (8 == 2*2*2)
  +-----+
 /       \
( 2 * 2 * 2 * 5 * 5) == (2**3 * 5**2) == 200  # lowest common denominator
          \       /
           +-----+
        (50 == 2*5*5)

Or much faster in python

import math
math.lcm(8, 50)       # 200 (lowest common multiple)
math.gcd(1920, 1080)  # 120 (greatest common divisor)