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Fractions
Intro
We use fractions everyday! If you buy a pizza and slice it up, each slice is a fraction of the whole
pie. Or, if you are baking a cake and are following a recipe, fractions are at work making sure the
correct amount of each ingredient is added.
a
A fraction is a number in the form , where a and b are integers and b ≠ 0.
b
The bottom number is called the denominator. The denominator tells you how many parts the
whole is divided into. The denominator of a fraction can never be zero.
The top number is called the numerator. The numerator tells you how many of those units you
have.
The diagram below shows you how the denominator and numerator are related.
1 3 3 8
, , , . You may
4 5 8 9
be asking, “How do I find the actual value of the fraction?” and the answer is by dividing the
numerator into the denominator. A fraction is actually a division problem in disguise!
Fractions were originally invented to show values less than 1; for example,
1
, I would divide 1 ÷ 4 which equals 0.25. Every
4
fraction can be converted into a decimal by dividing the numerator into the denominator.
For example, if I wanted to find the value of
When the numerator is less than the denominator, we name it a proper fraction,
1
Knowledge of number sense, concepts and operations.
Compare the relative values of fractions, decimals, percents and other real umbers.
1
Improper Fractions
Earlier we stated fractions were invented as a way to express values less than 1. However, there
are fractions which are larger than 1 as well! We call these improper fractions—fractions whose
numerator is larger than or equal to the denominator.
12
is and example of an improper fraction. If we were to divide this fraction our answer would be
7
greater than 1: 12 ÷ 7 = 1.7.
Mixed Numbers
A mixed number consists of a whole number and a fraction. Let’s look at our improper fraction
12
12
5
from above,
. This fraction can be expressed as a mixed number:
is equivalent to 1 (we
7
7
7
say, “one and five-sevenths”). Changing fractions to mixed numbers is usually not advised in
algebra because it may cause confusion. When working with improper fractions, unless
specifically stated in the problem, keep your answers as improper fractions.
One reason for the confusion is because, in algebra, when two quantities are written next to each
other with no symbols in between, multiplication is assumed. However, when using mixed
5
5
numbers addition is implied. For example, 1 means 1 + .
7
7
Conversion of Mixed Numbers
Mixed Number to Improper Fraction
Steps
5
1
7
+
x
1
1) Multiply the denominator times the whole number:
1x7=7
2) Add the result to the numerator: 7 + 5 = 12
3) Write the improper fraction:
12
(Note: the
7
denominator will always stay the same).
Knowledge of number sense, concepts and operations.
Compare the relative values of fractions, decimals, percents and other real umbers.
2
Improper Fraction to Mixed Number
Steps
14
5
1) Divide the denominator into the numerator, the integer
2
value becomes your whole number: 5 14 (remainder 4)
2) The remainder becomes your numerator: 2
4
(Note: The
5
denominator always stays the same).
Equivalent Fractions
Fractions that have the same value are called equivalent fractions. For example,
1
is equivalent
2
2
; to check whether two or more fractions are equivalent you can: 1) divide each fraction and
4
see if the decimals are the same or 2) if both the numerator and denominator increase by the
2
1
same amount across each fraction—notice is double (both the numerator and denominator
4
2
increase by 2).
to
3 9 27
, , ? To find our solution,
6 18 54
we can divide each fraction and compare their decimals—in this case, they are all 0.5. Or, we can
see if the numerators and denominators increase by the same amount—in this case, by
multiplying 3 each time. Therefore, these are equivalent fractions.
Here’s a second example: Are the following fractions equivalent,
Reciprocals
Two fractions whose product is 1 are reciprocals.
For example,
2
3
2 3 6
and are reciprocals because x = = 1.
3
2
3 2 6
To find the reciprocal of a whole number, simply put 1 over the whole number: the reciprocal of 3
1
is .
3
1
Knowledge of number sense, concepts and operations.
Compare the relative values of fractions, decimals, percents and other real umbers.
3