Chapters 3 & 4 FRACTIONS
A fraction is just a division problem.
Numerator
= Numerator ÷ Denominator
Denominator
A fraction is in LOWEST TERMS, also known as REDUCED or SIMPLEST FORM when the numerator and denominator have
no common factors.
6
3
is not in lowest terms,
is in lowest terms
10
5
A fraction can be put in reduced form by dividing both the numerator and the denominator by the GCF.
This can be done easily by putting both the numerator and denominator in prime-factored form and canceling the common factors
between the numerator and denominator.
6
10
2 3
2 5
3
5
Mixed Number - A number, such as
3
6 , consisting of an integer and a fraction. A mixed number is just the sum of a whole
5
number and a fraction.
Improper Fraction - A fraction in which the numerator is larger than the denominator. Converting from mixed number to
improper fraction:
6
3
5
6 5 3
5
33
5
Mixed numbers must be converted to improper fractions or decimals before doing ANY MULTIPLICATION OR
DIVISION OPERATIONS on them.
denominator numerator
A fraction can be converted into a decimal by dividing: Numerator ÷ Denominator
6
5 33
6
30
33
33 5
5
3
5
3
When ADDING or SUBTRACTING FRACTIONS that must have the same denominator, then you just add the
numerators and leave the denominator the same.
1 3 4
5 5 5
However, consider
1
6
3
8
The denominators, 6 and 8, are not the same, so we must find the LEAST COMMON DENOMINATOR to convert these fractions
into equivalent ones with the same denominator. The LCM of 6 and 8 is the SMALLEST NUMBER THAT BOTH 6 and 8 can go
into. Choose the larger denominator (which is 8 in this case) and start taking multiples until 6 can go into it.
Does 6 go into 8? NO Does 6 go into 8x2? NO Does 6 go into 8x3? Yes, 6 goes into 24. Therefore 24 is the LCM.
Multiply the numerator and denominator of each fraction by whatever it takes to get the LCM as the new denominator.
Then once you have the same denominators in each fraction, just add the numerators and leave the denominator the same.
1 4
6 4
3 3
8 3
4
24
9
24
13
24
Adding & Subtracting Mixed Fractions using
the Improper Fraction Method
convert them to Improper Fractions
then add them by finding the LCD and building up
each fraction to have the LCD.
then convert back to Mixed Fractions:
Example: What is 2 3/4 + 3 1/2 ?
Convert to Improper Fractions:
2 3/4 = 11/4
3 1/2 = 7/2
Common denominator of 4:
11
/4 stays as 11/4
7
/2 becomes 14/4
Now Add:
11
/4 + 14/4 = 25/4
Convert back to Mixed Fractions:
25
/4 = 6 ¼
Subtracting Mixed Fractions
Just follow the same method, but subtract instead of add:
Example: What is 15 3/4 - 8 5/6 ?
Convert to Improper Fractions:
15 3/4 = 63/4
8 5/6 = 53/6
Common denominator of 12:
63
/4 becomes 189/12
53
/6 becomes 106/12
Now Subtract:
189
/12 - 106/12 = 83/12
Convert back to Mixed Fractions:
83
/12 = 6 11/12
ADDING & SUBTRACTING MIXED NUMBERS
USING THE VERTICAL METHOD
ADDING MIXED NUMBERS
Step 1: Find the Least Common Denominator
Step 2: Find equivalent fractions with the LCD as
the new denominator
Step 3: Add the fractions and add the whole
numbers.
Step 4: Write your answer in lowest terms.
SUBTRACTING MIXED NUMBERS
The method is similar to that above, but after you
have found the LCD and built up equivalent fractions
and the numerator of the fraction you are
subtracting is larger than the fraction you are
subtracting from, you have to borrow.
5
12
2
1
3
3
Step 1: Find the Least Common Denominator.
LCD of 3 and 12 is 12.
Step 2: Find equivalent fractions with the LCD as
the new denominator
5
5
3
12
12
2 4
8
1
1
3 4
12
3
Step 3: Since numerator 5 is less than the
numerator 8, we must borrow, then subtract.
5
12
8
1
12
3
2
12 5
12 12
17
12
8
1
12
9
1
12
2
Step 4: Write your answer in lowest terms.
1
9
3 3
3
1
1
12
2 2 3
4
MULTIPLYING FRACTIONS
Multiplying Fractions uses a different rule. When multiplying fractions, you multiplying the numerators AND the denominators.
1
6
3
8
1 3
6 8
3
48
This fraction can simplified by canceling out common factors in the numerators and denominators (tops and bottoms)before
multiplying across. 6 can be rewritten as 2x3, and the 3’s can be cancelled out.
1
6
3
8
1 3
6 8
1 3
2 3 8
1
16
Remember, Mixed numbers must be converted to improper fractions or decimals before doing ANY MULTIPLICATION
OR DIVISION OPERATIONS on them.
1 2
6
2 3
4
4
1
2
6
2
3
9 20
2 3
3 3 2 2 5
2
3
30
1
30
DIVIDING FRACTIONS
The rule for dividing fractions is simple. Just take the RECIPROCAL(flip the top and bottom) of the DIVISOR (the fraction to the
right of the ÷ symbol) and multiply
1
6
3
8
1
6
8
3
1 8
6 3
1 2 2 2
2 3 3
4
9
A Complex Fraction is a fraction whose numerator and/or denominator contains a fraction. Treat it as a division problem.
1
3
1
6
6
8
3
8
1
6
8
3
1 8
6 3
1 2 2 2
2 3 3
4
9
POWERS
A power is a multiplication expression where the exponent tells you how many times to multiply a number (called the base) by
itself.
3
4
2
3
4
3
4
3 3
4 4
9
16