1. INTRODUCTION TO FRACTIONS
Math 081
CCBC Essex

2. Introduction to Fractions
A fraction represents the number of equal parts of a whole
Fraction = numerator (Top)
denominator (Bottom)
= numerator/denominator
Numerator = # of equal parts
Denominator = # of equal parts that make up a whole

3. Example: My wife and I ordered a large Papa John’s pizza
Example: My wife and I ordered a large Papa John’s pizza. The large pizza is cut into 8 (equal) slices. If I ate 3 slices, then I ate
3/8 of the pizza
3 Equal Parts
8 Parts make the whole

4. Types of Fractional Numbers

5. Converting Between Fraction Types
Any integer can be written as an improper fraction
Any improper fraction can be written as a mixed number
Any mixed number can be written as an improper fraction

6. Integer  Improper Fraction
The fraction bar also represents division
The denominator is the divisor
The numerator is the dividend
When you divide you get the quotient
To write an integer as a division problem, what do we divide a number by to get the number?
One n = n/1

7. Ex: Write 17 as an improper fraction
17 = 17 / ?
17 divided by what is 17? 1
Therefore, 17 = 17 / 1

8. Improper Fraction  Mixed Number
Denominator: tells us how many parts make up a whole
Numerator: tells us how many parts we have
How many wholes can we make out of the parts we have?
Divide the numerator by the denominator  the quotient is the whole part
How many parts do we have remaining?
The remainder (over the denominator) makes up the fraction part

9. Ex: Write 11/8 as a mixed number.
How many parts make up a whole? 8
Draw a whole with 8 parts:
How many parts do we have? 11
To represent 11/8 we must shade 11 parts . . .
But we only have 8 parts. Therefore, draw another whole with 8 parts . . .
Keep shading . . . 9 10 11
This is what 11/8 looks like.

10. Given the representation of 11/8, how many wholes are there? 1
Dividing 11 parts by 8 will tell us how many wholes we can make: 11/8 =
1 R ?
The remainder tells us how much of another whole we have left:
1 R 3
Since 8 parts make a whole, we have 3/8 left.

11. Mixed Number  Improper Fraction
Denominator: tells us how many parts make up a whole. Break down each whole into that many parts. How many parts do we get?
Multiply the whole number by the denominator.
Numerator: tells us how many parts we already have. How many parts do we now have in total?
Add the number of parts we get from breaking down the wholes to the number of parts we already have
Form the improper fraction:
# of parts
# of parts that make a whole
* The denominator ALWAYS stays the same when switching from a mixed number to an improper fraction.

12. Draw the mixed number
Looking at the fraction, how many parts make up a whole? 8
Chop each whole into 8 pieces.
How many parts do we now have? 8
+ 8
+ 5
= 8 * = 21
= parts from whole + original parts

14. Finding Equivalent Fractions
Equal fractions with different denominators are called equivalent fractions.
Ex: 6/8 and 3/4 are equivalent.

15. The Magic One
We can find equivalent fractions by using the Multiplication Property of 1: for any number a, a * 1 = 1 * a = a
We will just disguise the form of the one
Do you agree that 2/2 = 1?
How about 3/3 = 1?
4/4 = 1?
25/25 = 1? /17643 = 1?
1 has many different forms . . .
1 = n/n for any n not 0

16. Ex: Find another fraction equivalent to 1/3
1/3 = 1/3 * 1
We can write 1/3 many ways just be using One
= 1/3 * 2/2
= 2/6 or
1/3 = 1/3 * 1
= 1/3 * 3/3
= 3/9

17. Ex: Find a fraction equivalent to ½ but with a denominator of 8
1/2 = 1/2 * 1
We can write 1/2 many ways just be using One. We want a particular denominator – 8. What can we multiply 2 by to get 8?
= 1/2 * 4/4
Notice: 4
so choose the form of the One

18. Ex: Find a fraction equivalent to 2/3 but with a denominator 12
2/3 = 2/3 * 1
We can write 2/3 many ways just be using One. We want a particular denominator – 12. What can we multiply 3 by to get 12?
= 2/3 * 4/4 4
so choose the form of the One

19. Simplest Form of a Fraction
A fraction is in simplest form when there are no common factors in the numerator and the denominator.

20. Ex: Simplest Form Ex: 6/8 and 3/4 are equivalent
The fraction 6/8 is written in simplest form as 3/4 = = =
1 x
Magic one

21. Greatest Common Factor and Least Common Multiples GCF and LCM

22. What is the difference between a factor and a multiple?

23. Give me an example of a factor of 15

24. Give me an example of a multiple of 15

25. How would you find the GCF of 60 and 96?

26. There are actually 3 ways
There are actually 3 ways. You can use prime factorization, write out all the prime factors for each number or extract common prime numbers.

27. List the factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60 List the factors of 96 1,2,3,4,6,8,12,16,24,32,48,96 find the largest factor - 12

28. …or do prime factorization
…or do prime factorization. Circle all the primes the 2 numbers have in common and multiply one set of them to get your GCF. 96 60 48 2 2 30 24 2 15 2 12 2 3 5 6 2
2 x 2 x 3 = 12 2 3

29. This, to me, is the easiest way. Try it!
…or you can extract the common prime factors of 60 and 96 and form a product to give the GCF.
This, to me, is the easiest way. Try it! 2
60, 96
2 x 2 x 3 = 12 2
30, 48 3
15, 24
5, 8

30. Find the GCF of 36, 24, 144 and 96

31. 96 24 36 48 2 2 12 18 2 24 2 6 9 2 2 12 2 3 2 3 3 6 2
144 2 3 12 12 3 4
2 x 2 x 3 = 12 3 4 2 2 2 2

32. 2
36, 24, 144, 96
2 x 2 x 3 = 12 2
18, 12, 72, 48 3
9, 6, 36, 24 3 2 12 8

33. There are 3 ways to find the LCM as well
There are 3 ways to find the LCM as well. You can list the multiples of the numbers, do prime factorization, or extract the prime factors from either one or both numbers. Find the LCM of 12 and 18

34. Multiples of 12 are… 12,24,36,48,60,72,…. Multiples of 18 are… 18,36,54,72,90,108,… The smallest multiple the 2 numbers have in common is the least common multiple.

35. …or do prime factorization
…or do prime factorization. Write down the number they have in common only once, then write down the leftover numbers. Multiply them all together. 12 18 9 4 2 3 3 3 2 2
Numbers in common are 2 and 3
Leftover numbers are 2 and 3
2 x 3 x 2 x 3 = 36

36. Remember…the numbers you pull out have to be prime numbers!2
12, 18 2
6, 9 3
3, 9 3
1, 3 1 1
2 x 2 x 3 x 3 = 36

37. Find the LCM of 35, 420 and 245

38. 245 35
420 42 5 49 10 5 7 5 7 7 2 6 7 2 3
Numbers they have in common: 5 and 7
Leftover numbers: 2, 3, 2, 7
Multiply them all together: 5 x 7 x 2 x 3 x 2 x 7 = 2940

39. primes 5
35, 420, 245 7
7, 84, 49 7
1, 12, 7 3
1, 12, 1 2
1, 4, 1 2
1, 2, 1
1, 1, 1
5 x 7 x 7 x 3 x 2 x 2 = 2940

40. Find the GCF and LCM of 28 and 63

41.
GCF = 7
LCM = 7 x 4 x 9
= 252

42. Find the GCF and LCM of 12 and 13

43.
GCF = 1
LCM = 1 x 12 x 13
= 156

44. Find the GCF and LCM of 27, 21, 33 and 15

45.
GCF = 3
LCM = 3 x 9 x 7 x 11 x 5
= 10,395

46. Ex: Write 12/42 in simplest form
First prime factor the numerator and the denominator:
12 = 2 x 2 x 3 and 42 = 2 x 3 x 7
Simplify = = =
1 x 1 x =
Notice: 2 x 3 = 6 = GCF(12, 42)
 factoring (dividing) out the GCF will simplify the fraction

47. Ex: Write 7/28 in simplest form
What is the GCF(7, 28)?
Hint: prime factor 7 = 7
prime factor 28 = 2 x 2 x 7
= 7 = = =
1 x =
Dividing out the GCF from the numerator and denominator simplifies the fraction.

48. Ex: Write 27/56 in simplest form
What is the GCF(27, 56)?
Hint: prime factor 27 = 3 x 3 x 3
prime factor 56 = 2 x 2 x 2 x 7
= 1
There is no common factor to the numerator and denominator (other than 1)
Therefore, 27/56 is in simplest form.