Download Fraction Basics

Fraction Basics
Simplifying Fractions
Knowing your multiplication facts and/or divisibility rules helps when figuring
out what both the numerator and denominator of a fraction have in common
and then reducing it to lowest terms.
Example 1:
24
24 ÷ 2 12 12 ÷ 2
6
6÷2
3
=
=
=
=
=
=
56
56 ÷ 2 28 28 ÷ 2 14 14 ÷ 2 7
24
24 ÷ 8 3
Example 2:
=
=
€
€ 56 €56 ÷ 8€ 7
€
€
€
Notice that the second example used knowledge of the multiplication facts
to simplify
in less steps than the first example but that both
€
€the fraction
€
answers are the same.
Equivalent Fractions
To find the missing number, first figure out what the given numerator or
denominator was multiplied by to get its corresponding numerator or
denominator, then multiply that number to get the missing number.
Example:
5
5×6
5
30
=

=

=
12 72
12 × 6 72
12 72
Always remember that whatever you multiply the top (numerator) by you
must€multiply
€ the€bottom €(denominator)
€
€ by!
Prime Factorization
Use a factor tree to break down a number into its prime numbers, then
rewrite the prime factorization using exponents in ascending order.
Example:
90 = 2 × 32 × 5
90
9
3
10
3
2
The exponent tells how many of the
base number are multiplied together.
3 × 3 = 32
5 × 5 × 5 = 53
2 × 2 × 2 × 2 × 2 = 25
5
Writing Improper Fractions as Mixed Numbers
What is nice is that the fraction tells you how to do this as it is another
form for writing a division problem.
Example:
€25
6
4
25
25
1
 6) 25  6) 25 
=4
6
6
6
-24
1
€ numerator
€
divided by
denominator
€
Notice that the mixed
number is written with
the quotient as the whole
number and with the
remainder over the
divisor as the fraction.
€
Writing Mixed Numbers as Improper Fractions
Multiply the whole number by the denominator, add the numerator, then put
it over the denominator.
Example:
€
6
2
2 20
 6×3+2  6 =
3
3 3
€
GCF (Greatest Common Factor)
GCF and LCM are often confused with each other. If you focus on the last
word of each, factor and multiple, then you will remember that factors are a
part of a product (smaller) and multiples are more of a product (bigger).
There are three different methods to finding the GCF, but the Cake Method
is the easiest especially when three numbers are involved.
Method 1: List the factors.
12 = 1, 2, 3, 4, 6, 12
16 = 1, 2, 4, 8, 16
GCF = 4
Notice they have both 2 and 4 in
common, but 4 is the greatest.
Method 2: Prime Factorization
12
16
3
4
2
4
2
2
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
GCF = 2 × 2 = 4
4
2 2
2
Method 3: Cake Method
It is a kind of upside-down division where you divide both
numbers by the same common number and keep doing so until
the only number in common is 1.
2 | 12 16
2 | 6 8
3 4
Only multiply
the numbers
on the side to
get the GCF.
GCF = 2 × 2 = 4
There is nothing in common
with 3 and 4 except 1.
LCM (Least Common Multiple)
There are also three different methods to finding the LCM, but in this case
not one method is better than the rest. It all depends on the numbers
involved. Listing the Multiples is easiest for smaller numbers, Prime
Factorization is easiest for larger numbers, and the Cake Method is the
easiest when there are only two numbers involved because it doesn’t work
with three.
Method 1: List the Multiples
12 = 12, 24, 36
18 = 18, 36
LCM = 36
In truth, I really don’t list out all of the multiples. I start with
the larger of the two numbers and ask myself if the smaller
goes into it evenly. If that doesn’t work, then I work through
its multiples.
18  Does 12 go evenly into 18? No
18 × 2 = 36  Does 12 go evenly into 36? Yes, LCM = 36.
Method 2: Prime Factorization
Use the factor tree, then circle all of the prime numbers with
the highest prime numbers.
12 = 22 x 3
18 = 2 × 32
LCM = 22 x 32 = 36
Method 3: Cake Method
3 | 12 18
2| 4 6
2 3
LCM = 3 × 2 × 2 × 3 = 36
Notice for the LCM that all of the numbers outside of the
upside-down division signs are multiplied together.
Comparing Fractions
Again, this is a case where the fractions dictate which method to use is
best.
Method 1: Make Equivalent Fractions
9
16
5
9

8
16
5 × 2 10
9
5

<
=
8 × 2 16
16 8
Method 2: Cross Multiply
€
€
€
4
9
Ordering
Fractions
€
€
€
× 9 €20
2
4

5
9
€ ×5
2
4

5
9
€
€
€
€ 18
2
4
2

>
5
9
5
€
€
Give all of the fractions a common denominator, then order them. LCM is
very helpful here.
1. The LCM of 6, 12, and 4 is 12, so the
5 7
3
common denominator is 12.
Example:
6 12 4
2. Make equivalent fractions.
  
3. Now it is easy to see which fraction is the
5×2 7
3× 3
least and which is the greatest.
6 × 2 12 4 × 3
€ € €
4. Order the fractions using the original
  
fractions.
10 7
9
12 12 12
€
€ €
7 3 5
Therefore the order from least to greatest is
12 4 6
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