ADDING FRACTIONS

Rule for Adding Fractions:

1) In order to add fractions, their denominators (bottom
numbers) must be the same.

2) Once the fractions have been rewritten (if necessary) with a
“least common denominator ”, add them by
ADDING the NUMERATORS (top numbers) and KEEPING THE
DENOMINATOR the SAME.

Example1

Can be 6/8 reduced ? Yes.
 

Example

The denominators, 4 and 3, are not the same, so we have to find the
LCD (Least Common Denominator) in order to rewrite these fractions
into equivalent fractions with the same denominator.
The LCD is the “smallest box that both 4 and 3 will go into.”

ADDING FRACTIONS WITH DIFFERENT DENOMINATORS
FINDING LCD METHOD 1:
1) Start making multiples of the larger denominator until the smaller
denominator can go into it.

4·1= 4 Does 3 go into 4? No
4·2= 8 Does 3 go into 8? No
4·3= 12 Does 3 go into 12? YES…. STOP
12 is the LCD of 4 and 3

Now rewrite each fraction with 12 as the denominator, then add.

METHOD 2:
Example 8:

1) Write the prime factorization of each denominator .
21 = 3·7
18 = 2·3 ·3

2) Circle all the prime factors of the first denominator.
21= 3 · 7
18= 2 · 3 · 3

Then circle all the prime factors of the second denominator that DON’T
appear in the first denominator's prime factors. Scratch out factors
that are in the first denominator.
Notice that 21 = 3·7, which has a 3 in the prime factorization. The
second denominator is 18 = 2 ·3 ·3, which has two 3’s. One 3
already appeared in the prime factorization of 21.

The LCD is the circled factors. LCD = 3·7·2 ·3=126

3) Rewrite each fraction into an equivalent fraction with the LCD as the
denominator

4) Now perform the subtraction by subtracting the numerators and
leaving the denominator the same.

Example 5: Add -5 + ¼

Example 6: Add

Example 7
Subtract:

Are the denominators, y and 18, the same?
Then, find the LCD of y and 18.

But we don’t know what y is, so how could we know if it goes into
18?

When you have denominators where one is a variable term and the
other is a constant, term find the Least Common Multiple of the
coefficient and the constant , and then find the Least Common
Multiple of the variables.

The coefficient of y is 1. The other denominator is just 18.
What is the LCM of 1 and 18?
Remember, LCM means what’s the smallest number that both 1 and
18 will go into?
LCM = ____

However, the Least Common Denominator will have to include the
the LCM of the variables. Since the other denominator doesn’t have
a variable, the LCD of the variables is just y.
So the LCD is _____
Now rewrite each fraction so that the denominator is ____

The new ex pression which now has like terms is:

Example
Subtract:

Are the denominators, 3y and 6y², the same?
Then, find the LCD of 3y and 6y².

The coefficient of 3y is 3. The coefficient of 6y² is 6.
What is the LCM of 3 and 6?
Remember, LCM means what’s the smallest number that both 3 and
6 will go into?
LCM = ____

However, the Least Common Denominator will have to include the
the LCM of the variables. What’s the LCM of y and y²? _____
So the LCD is _____
Now rewrite each fraction so that the denominator is ____

The new expression which now has like terms is:

Comparing Fractions

To compare fractions with different denominators, rewrite the
fractions as equivalent fractions with the least common
denominator.

Example 10
Which fraction is larger?5/6 or 7/8

What is the LCD of 6 and 8?

METHOD 1:
1)Start taking multiples of 8.
8·1=8 Does 6 go into 8? No
8·2=16 Does 6 go into 16? No
8·3=24 Does 6 go into 24? Yes. LCD = 24
2) Rewrite the fractions as equivalent fractions with the least
common denominator.

Which one is bigger?

4.5 MULTIPLYING AND DIVIDING MIXED NUMBERS

A proper fraction is a fraction whose numerator is smaller than the
denominator.
An improper fraction is a fraction whose numerator is larger than
the denominator.

Mixed Numbers

A mixed number is the sum of a whole number and a proper
fraction. The sum is implied because the plus sign is often not
included.

When multiplying or dividing mixed numbers, you must ALWAYS
convert them to improper fractions first.

How?
First, Multiplying the whole number by the denominator
of the fraction.

5·6=30 Then add the numerator, 1.
30 + 1 = 31

Now this number is your new numerator. Keep the denominator
the same.

NOW DO SC 1 on p. 262

HOMEWORK:

Sec. 4.4 #5-15 odd, #21-87 ETP,
Sec. 4.5 #17-77 odd
Ch. 4 Review p.297-298 #49-79
odd