Download algebra review:fractions

Algebra Review: Fractions
ALGEBRA REVIEW: FRACTIONS
DEFINITION
A fraction is a numerical value described as one value divided by
another.
Ex: 1 ÷ 3 = 1/3
The three parts of a fraction are the numerator, the denominator,
and the division symbol.
Numerator
Denominator
TYPES OF
FRACTIONS
Division symbol
Simple. Both numerator and denominator are integers.
1
57
2
212
Complex. There is a fraction or decimal within the fraction.
4.3
1/3
7
8
5
2/5
Compound. A whole number is combined with a simple fraction.
3
12
7
5
13
Improper. A simple fraction with a numerator that is greater than
the denominator.
4
5
2
WRITING
DECIMALS AS
FRACTIONS
17
13
Terminating decimals and complex fractions. Multiply top and
bottom by 10 until there is no longer a decimal. (See Multiplying
Fractions).
.57
1
Ex: 0.57 =
.57
1
×
10
10
=
5.7
10
5.7
10
×
10
10
=
57
100
1
Algebra Review: Fractions
Infinitely repeating decimals. Follow the steps in the example
below.
Ex: Write 0.272727 . . . as a fraction.
Pick a letter. I’ll use ‘n'.
Let n = 0.2727272727 . . .
This decimal breaks up to a repeating set of two digits
(0. 27 27 27 27 27 . . . ) so I’ll multiply both sides of my equation
by a one followed by two zeros, one hundred.
100n = 27.2727272727 . . .
Subtracting my original equation, I get
100n = 27.27272727 . . .
-100n = 00.27272727 . . .
99n = 27
Divide both sides by 99.
n =
27
99
27
99
0.27272727 . . . =
Non-terminating, non-repeating decimals. There is no way to write
these as exact fractions.
CONVERTING
FRACTIONS
Improper to Compound. Divide the numerator by the
denominator. Bring the answer out to the front and leave the
remainder in the fraction.
Ex:
17
13
is the same as 17 ÷ 13.
17 ÷ 13 = 1 R4
17
13
= 1
4
13
2
Algebra Review: Fractions
Compound to Improper. Multiply the outside number by the
denominator, and add that number to the numerator.
Ex:
2
1
2
=
(2 × 2) + 1
2
=
4+1
2
=
5
2
Complex fractions. Rewrite the fraction as a division problem.
Follow the steps covered in Dividing Fractions.
Ex: 1/3 = 1 ÷ 5 See Dividing Fractions.
5
3
REDUCING
SIMPLE
FRACTIONS
Factor the numerator and the denominator. If they have a common
factor, then it can be removed from the fraction.
Ex:
27
3×3×3
3
=
=
99
3 × 3 × 11
11
This applies to multiplication only.
Ex:
ADDING AND
SUBTRACTING
FRACTIONS
3
3+x
CANNOT be simplified to
1
x
Common denominators. Fractions can only be added or subtracted
if they have the same denominator. For example, (1/2 + 1/3) cannot
be simplified as is, but (3/6 + 2/6) can.
NOTE: This is the same sort of idea as combining like terms.
(a + b) is as simplified as it can get, but
(a + 2a) can be simplified to (3a).
Finding a common denominator.
First, find the least common multiple for the denominators.
Ex:
3
5
+
6
8
Multiples of 8: 8, 16, 24, 32, 40
Multiples of 6: 6, 12, 18, 24, 30
24 is the smallest number that is divisible by both 8 and 6.
3
Algebra Review: Fractions
Second, change both fractions so that the least common
multiple is the denominator of both. This is the opposite of
reducing fractions.
Ex:
5
4
20
×
=
6
4
24
(See Multiplying Fractions.)
3
3
9
×
=
8
3
24
So 5 + 3
6
8
becomes 20 + 9 = 29
24
24
24
The same steps can be used for subtraction of fractions. The
normal rules of addition and subtraction apply.
MULTIPLYING
FRACTIONS
To multiply fractions, multiply straight across the top and bottom, as
shown in the example below.
Ex:
DIVIDING
FRACTIONS
5
5
5×5
25
0×0
=
=
6
7
6×7
42
Reciprocals. The reciprocal of a fraction is the fraction flipped.
Ex: The reciprocal of
2
7
is
7
2
Dividing fractions. Dividing by a fraction is the same as multiplying
by the reciprocal of the fraction.
Ex:
Ex:
RATIONAL
EXPRESSIONS
5
2
5
7
35
÷
=
×
=
6
7
6
2
12
1
1
1
1
÷ 5 =
×
=
3
3
5
15
The rules of fractions apply even to more complicated rational
expressions like
3x2 + 4xy + y3 + 7
x4 - 5y7
4