Fractions
From wikinotes
Components
Fractions
numerator ------------- denominatorRatios/Rates
Ratios are just another way of expressing fractions.
ex:34:3
or34 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)
- find the lowest number divisible by both
4
and6
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 byYou can use the prime factorization of 2x numbers to find the lowest common denominator.
8 50 | \ | \ 2 4 5 10 | \ | \ 2 2 2 5If 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)