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Section 4.1 – Least Common Multiples
The Least Common Multiple (or LCM) of two natural numbers is the
smallest number that is a multiple of both numbers.
Method 1: Using Cuisenaire Rods
Build rods of lengths a and b until they create the same length train.
LCM(3, 4)
Method 2: Using a List
Example: Find the LCM of 3 and 5.
Example: Find the LCM of 4 and 12.
Example: Find the LCM of 4, 10, and 15.
Method 3: Using Prime Factorizations
Find the prime factorization of each number. Take each prime factor that
occurs in either number. Use the largest exponent on each factor.
Example: Find the LCM of 18 and 20.
Example: Find the LCM of 12 and 60.
Example: Find the LCM of 27, 90, and 84
Example: Find the LCM of 7x and 7xy
Example: Find the LCM of 10x4 and 5x3
Example: Find the LCM of 6ab2 and a3b
Example: A radio station gave away a T-shirt for every 12th and 13th caller.
Every 20th caller received a pair of concert tickets. Which caller was the first
to receive a T-shirt and a pair of concert tickets?
Section 4.2 – Addition of Fractions, Order, and Applications
Adding fractions is just like combining like terms, but instead of having
2x + 3x = 5x, we will have something like 2 eighths + 3 eighths = 5 eighths.
To add two fractions that have the same denominator, just add the
numerators and put the sum over the original denominator.
For example: 1  2  1 2  3
7 7
7
7
Examples: Add and simplify the answer
a) 11  4 
3
3
b) 2  7 
9
9
c) 5  6 
x x
Example: Simplify by combining like terms:
a) 2 x  5 x
3
3
b) 5 a  25 a  7 a
6
52
4
To add two fractions that have different denominators, find the LCM of
the denominators (called the LCD), convert both fractions so they both
have the LCD as their denominator (this involves multiplying by 1), and
then add the numerators.
Examples: Add and simplify the answer
a) 3  1 
4 6
b) 3  3 
5 8
c) 5  4 
6 9
d) 2  2 
7
e) 11  6 
21 15
Ordering Fractions:
To compare fractions, change both fractions so that they have the same
denominator, then compare numerators.
Examples: Fill in the blank with > or < or =
a) 5
8
______
b) 3
8
______
c) 87
100
d) 5
14
2
3
______
______
2
7
9
10
8
21
Example: Order the following fractions from smallest to largest
5
6
19
21
11
14
Applications:
Example: The label on a bottle of juice says that ¾ of the bottle consists of
apple juice, 1/6 of the bottle consists of cherry juice, and the rest is water.
What fraction of the bottle is juice?
Example: Jane took 1/5 gram of ibuprofen before lunch and ½ gram after
lunch. How much did she take altogether?
Example: A recipe calls for ½ quart of buttermilk, 1/3 quart of skim milk,
and 1/16 quart of oil. How many quarts of liquid ingredients does the recipe
call for?
Section 4.3 – Subtraction of Fractions, Equations, and Applications
To subtract two fractions that have the same denominator, just subtract
the numerators and put the difference over the original denominator.
For example: 4  1  4 1  3
7 7
7
7
Examples: Subtract and simplify the answer
a) 10
9

2 
9
b) 2
5

9 
5
c) 13
4x

1 
4x
To subtract two fractions that have different denominators, find the
LCM of the denominators (called the LCD), convert both fractions so
they both have the LCD as their denominator (this involves multiplying
by 1), and then subtract the numerators.
Examples: Subtract and simplify the answer
a) 3
5

1 
2
b) 7
12

5 
6
c) 11  5 
28 16
d) 9 x  3 x 
5
10
Solving Equations: Solve just like we did before…
Examples: Solve the following equations
a) x  2  1
5
5
b) x

c) 11
32
4  5
7
14

x 7
8
Applications:
Example: As part of an exercise program, Hugo walks 7/8 miles each day.
He has already walked 1/3 mile today. How much farther should Hugo walk?
Example: Amber added 1/3 quart of 2-cycle oil to a fuel mixture for her lawn
mower. She then noticed that the owner’s manual indicates ½ quart should
have been added. How much more 2-cycle oil should Amber add to the
mixture?
Example: The label on a bottle of juice says that ¾ of the bottle consists of
apple juice, 1/6 of the bottle consists of cherry juice, and the rest is water.
What fraction of the bottle is water?
Example: A new Chevy Prizm costs $14,400. Pam will pay ½ of the cost,
Sam will pay ¼ of the cost, Jan will pay 1/6 of the cost, and Nan will pay the
rest. What fractional part will Nan pay? How much will Nan pay?
Section 4.4–Solving Equations
Examples: Solve using the multiplication and addition principles
a) 7 x  6  22
b) 31  3x  5
c) 2 x  3  3
5
5
10
d) 1  2 t  4
5
5
e) 7  5  3 y
2
f) 17  2 x  3
7
2
2
Using the Multiplication Principle to Clear Fractions:
Multiply each term by the LCD.
Examples: Solve by clearing fractions
a) 1 x  1  2
3
6
3
b) 1 x  4  3
5
3
4