Section 2.2 Simplifying Fractions (113)
Objectives: Write a number as product of prime factors, reduce a fraction to lowest terms, determine if two fractions are equal
2.2.1 Writing a Number as a Product of Prime Factors
Definition: A prime number is a number > 1 that cannot be evenly divided except by itself and 1.
If you are curious, prime numbers are not new. The Greek mathematician Eratosthenes divested a method of locating prime numbers called the Sieve of Eratosthenes. The method is still used computers. Look it up if you care.
The first 15 prime numbers are { 2, 3, 5, 7, 11, 17, 23, 29, 31, 37, 41, 43, 47} Normally this is sufficient
A composite number (non-prime number) can be divided evenly by numbers other than itself and 1 (not necessarily unique) Examples: 14 = 2 X 7 16 = 2 X 8 or 4 X 4
Remember the factors are numbers that are multiplied together.
Divisibility Tests
A number can be divided by 2 if the last digit is 0, 2, 4, 6, or 8
A number can be divided by 3 if the sum of the digits is divisible by 3
(e.g. 2304 [2+3+0+4=9, 9 divisible by 3)
A number can be divided by 5 if the last digit is 0 or 5
You will discover later that normally we would like to find prime factors of a number.
TE1 Write each whole number as a product of primes. a) 28 2 2 X 7
b) 40 2 3 X 5 c) 300 2 2 X 3 X 5 2
The method I suggest to break a number into a product of primes is divide by 2s until you can’t, then3s, then 5s, etc. until you get down to 1. The power of each prime is the number of times you divided in the fray 2 ) 300
2 ) 150
3 ) 75
5 ) 25
5 ) 5 There are two 2s, one 3, and two 5s
1
PP1 (114) Write each whole number as a product of primes: a) 18 2 X 3 2 b) 72 2 3 X 3 2 c) 400 2 4 X 5 2
Fundamental Theorem of Arithmetic: Every composite number can be written in exactly one way as a product of prime numbers. (Normally written in ascending values of the primes)
2.2.2 Reducing a Fraction to Lowest Terms
Note: 5 + 2 and 3 + 4 are two ways to write the same number. We say they are
EQUIVALENT, because they are equal.
Fractions can also be written the same way. An example is that 15/18 and 10/12 are equal.
What we really need to do is to make the numerator and denominator smaller if possible,
SIMPLIFYING THE FRACTION. You make the terms as small as you can. It is called the
SIMPLEST FORM.

Remember any number divided by itself is 1. (e.g. 3/3, 17/17 )
If b and c are not 0, then a/b = (a X c)/(b X c) 1/3 = (1 X 4)/(3 X 4) = 4/12
Look at it in reverse. 14/21 = (2 X 7)/(2 X 7) factor 7 from numerator and denominator, the result is 2/3
A fraction is called SIMPLIFIED, REDUCED, or IN LOWEST TERMS if the numerator and denominator have only 1 as a common factor. Starting now, and that includes Test 1, I expect all fractions to be simplified.
TE2 Simplify (write in lowest terms. a) 18/27 = (2 X 3 2 )/(3 3 ) 2//3
b) 48/56 = (2 4 X 3)/(2 3 X 7) = (2 X 3)/7 = 6/7
PP2 (116) Simplify by dividing out common factors. a) 30/42 = (2 X 3 X 5)/(2 X 3 X 7) = 5/7 b) 60/132 = (2 2 X 3 X 5)/(2 2 X 3 X 11) = 5/11
TE3 Simplify the fractions a) 40/48 = (2 3 X 5)/(2 4 X 3) = 5/(2 X 3) = 5/6 b) 60/102 = (2 2 X 3 X 5)/(2 X 3 X 17) = (2 X 5)/17 = 10/17
PP3 (116) Simplify the fractions a) 120/135 = (2 3 X 3 X 5)/(3 3 X 5) = 2 3 /3 2 = 8/9
2.2.3 Determining Whether Two Fractions are Equal (117)
Equality Test: If two fractions ( a/b and c/d [b and d ≠ 0] ) are equal then a X d = b X c
TE4 Are these fractions equal? a) 15/35 and 3/7 15 X 7 =? 35 X 3 105 = 105 yes, they are equal b) 7/12 and 86/144 7 X 144 =? 12 X 86 1008 =? 1032 no, they are not equal
PE4 (117) Test whether the following fractions are equal? a) 84/108 and 7/9 84 X 9 =? 108 X 7 756 =? 756 yes, they are equal b) 3/7 =? 79/182 3 X 182 =? 7 X 79 546 =? 553 no, they are not equal