BeginningAlgebra

Addition/Subtractions of Fractions

Addition & Subtraction

with Fractions

Add/Subtract Fractions

Fractions

Number Line

.LCD/ Fractions Sums

Primes/Factors

Prime Numbers and Factoring

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...

Prime NumbersPrime Numbers::

Counting Numbers < 121 not divisible by 2, 3, 5, 7 are Prime NumbersPrime Numbers.

Positive Integer or Natural Number divisible divisible only by itself and 11.

Prime NumberPrime Number:

Prime Numbers and Factoring

= 124 3 = 124 3Factors Product

Factoring

Multiplying

.

Prime Numbers and Factoring

Factors

Prime Factors

Exponent Form

60 = 4 15

60

= 2 32 5

32

2= 560

Positive Integer or Natural Number divisible divisible only by itself and 11.

Prime NumberPrime Number:

630 =

Prime Numbers and Factoring

63 10 Factors

Prime Factors

Exponent Form

Factors9)(7 (2 5)

2 3 537

52

3 72

All divisorsdivisors of 630:

630, 315, 210, 126, 105, 90, 70, 63, 45, 42, 35, 30, 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21

630 =

630 =

630 =

Note vertical pairsNote vertical pairs::

… - 5 - 4 - 3 - 2 -1 0 1 2 3 4 5 6 7 8 9 …

NUMBER LINE

OriginNegative numbers Positive numbers

Negative direction Positive direction

• … - 5 - 4 - 3 - 2 -1 0 1 2 3 4 5 6 7 8 9 …

• • • •

- 4.5 6.54.041-1 4

1

Coordinates of points

RECIPROCALSnumbers with product of 1numbers with product of 1

5 = 115

a = 11a

-23

- 32

= 1

cd

dc

= 1

.

Denominator b

English words Math symbols

… - 5 - 4 - 3 - 2 -1 0 1 2 3 4 5 6 7 8 9 …

.

Numerator aNumerator

Denominator

FRACTIONS

ba

728

3- 5

2- 8

65

81

43103

535

4

104

2

4

3 = 14

4

0 = 12

2

2

1

= 18

808

1

8

2

8

3

8

4

8

5

8

6

8

7

ab

a nb n

=Equivalent Fractions

FRACTIONS

Reduce Fraction to Lowest Terms

Prime Factors630240

= 2 3 537

2 2 322 5

= 2 3 537

2 2 322 5Divide by 2 3 5

= 37 222

= 218

Reduce fraction

Reduced improperimproper fraction630240

ba c

d or a

bcd

Multiply Fractions

Check the result to see if it will reduce to simpler formreduce to simpler form.

ac a c

Multiply numerators numerators timestimes numerators numerators

= bdb d

and denominatorsdenominators times denominatorsdenominators

94 7

8

Multiply Fractions

Check the result to see if it will reduce to simpler formreduce to simpler form.

or 49

78

1

2

[If numerators have common factors with denominators, factor them out and reducereduce before you multiply.]

7 1 7Multiply numerators numerators timestimes numerators numerators

18=

9 2and denominatorsdenominators times denominatorsdenominators

.

Check the result to see if it will reduce to simpler formreduce to simpler form.

just add the numeratorsadd the numerators.

a + b=

To add fractions with common denominatorscommon denominators,

ca b

c+ c

Add and Subtract Fractions:

= 2 + 45

65=

52 4

5+1.

2.32 x

3– = 2 – x3

In algebra some of the factors may be lettersletters, or variables, which must be treated as prime factors. Letters will be left in the result as “literal sums”.

Add and Subtract Fractions:

Prime Factors

LCD

LCDLCD: LeastLeast CollectionCollection of Prime FactorsPrime Factors.

32 7

Least Common Denominator or LCD

42 =

Prime Factors 22 2 324 =

168 = 22 2 3 7

24

42

Add and Subtract Fractions:

ba c

d+

If the fractions do do notnot have common denominators have common denominators

= ab

dd

cd

bb

+

RAISERAISE all fractions to have common denominatorscommon denominators,[Factor the denominators and find the LCDLCD. ]

ad + bcbd

missing factorsmissing factors

= ad + bcbd

then add the resulting numeratorsadd the resulting numerators.

Check the result to see if it will reduce to simpler formreduce to simpler form.

Add and Subtract Fractions:

=241

+ 425

22 2 31

+5 32 7

+ 22 2 3 75

22 2 3 71

569

=

22 2 3 727= =

7missing factormissing factor

22missing factorsmissing factors

7 + 20 22 2 3 7=

9

1

Adding mixed numbers – integers and fractionsintegers and fractions:

20372

=3 · 72 - 7( 2 · 2) + 5 · 3

2 · 2 · 2 · 3 · 3

missing factor 1

missing factor 2missing factor 3

72 = 2 · 2 · 2 · 3 · 3

18

24

LCD

missing factor 1

missing factor 2

missing factor 3

1 ·72 = 2 · 2 · 2 · 3 · 3

24

18

LCD718

+ 524

=3

216 - 28 + 15 2 · 2 · 2 · 3 · 3

=

718

+ 524

=3

THEEND