Download Pre-Algebra Notes – Unit Five: Rational Numbers and Equations

Pre-Algebra Notes – Unit Five: Rational Numbers and Equations
Rational Numbers
Rational numbers are numbers that can be written as a quotient of two integers.
Since decimals are special fractions, all the rules we learn for fractions will work for decimals.
The only difference is the denominators for decimals are powers of 10; i.e., 101, 102, 103, 104,
etc.... Students normally think of powers of 10 in standard form: 10, 100, 1000, 10,000, etc.
In a decimal, the numerator is the number to the right of the decimal point. The denominator is
not written, but is implied by the number of digits to the right of the decimal point. The number
of digits to the right of the decimal point is the same as the number of zeros in the power of 10:
10, 100, 1000, 10,000…
Therefore, one place is tenths, two places are hundredths, and three places are thousandths.
Examples:
56
100
532
3 places →
1000
2
1 place → 3
10
1) 0.56
2 places
2) 0.532
3) 3.2
→
The correct way to say a decimal numeral is to:
1) Forget the decimal point (if it is less than one).
2) Say the number.
3) Then say its denominator and add the suffix “ths”.
Examples:
1)
2)
3)
4)
0.53
0.702
0.2
5.63
Fifty-three hundredths
Seven hundred two thousandths
Two tenths
Five and sixty-three hundredths
When there are numbers on both sides of the decimal point, the decimal point is read as “and”.
You say the number on the left side of the decimal point, and then the decimal point is read as
“and”. You then say the number on the right side with its denominator.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 1 of 27
Examples:
1) Write 15.203 in word form.
Fifteen and two hundred three thousandths
2) Write 7.0483 in word form.
Seven and four hundred eighty-three ten-thousandths
3) Write 247.45 in word form.
Two hundred forty-seven and forty-five hundredths
Converting Fractions to Decimals:
Terminating and Repeating Decimals
Syllabus Objective: (2.22) The student will write rational numbers in equivalent forms. (2.3)
the student will translate among different forms of rational numbers.
CCSS 8.NS.1-2: Understand informally that every number has a decimal expansion; show
that the decimal expansion of a rational number repeats eventually or terminates.
a
(quotient of two integers), will
b
either be a terminating or repeating decimal. A terminating decimal has a finite number of
decimal places; you will obtain a remainder of zero. A repeating decimal has a digit or a block
of digits that repeat without end.
A rational number, a number that can be written in the form of
One way to convert fractions to decimals is by making equivalent fractions.
Example:
1
to a decimal.
2
Since a decimal is a fraction whose denominator is a power of 10, look for
a power of 10 that 2 will divide into evenly.
Convert
1 5

2 10
Since the denominator is 10, we need only one digit to the right of the
decimal point, and the answer is 0.5.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 2 of 27
Example:
3
to a decimal.
4
Again, since a decimal is a fraction whose denominator is a power of 10,
we look for powers of 10 that the denominator will divide into evenly. 4
won’t go into 10, but 4 will go into 100 evenly.
Convert
3 75

4 100
Since the denominator is 100, we need two digits to the right of the
decimal point, and the answer is 0.75.
There are denominators that will never divide into any power of 10 evenly. Since that happens,
we look for an alternative way of converting fractions to decimals. Could you recognize numbers
that are not factors of powers of ten? Using your Rules of Divisibility, factors of powers of ten
can only have prime factors of 2 or 5. That would mean 12, whose prime factors are 2 and 3,
would not be a factor of a power of ten. That means that 12 will never divide into a power of 10
5
evenly. For example, a fraction such as
will not terminate – it will be a repeating decimal.
12
Not all fractions can be written with a power of 10 as the denominator. We need to look at
another way to convert a fraction to a decimal: divide the numerator by the denominator.
Example:
3
to a decimal.
8
This could be done by equivalent fractions since the only prime factor of 8
3
3 125 375
 

is 2.
8
8 125 1000
Convert
However, it could also be done by division.
0.375
8 3.000
Doing this division problem, we get 0.375 as the equivalent decimal.
Example:
5
to a decimal.
12
This could not be done by equivalent fractions since one of the factors of
12 is 3. We can still convert it to a decimal by division.
Convert
0.41666...
12 5.00000
Six is repeating, so we can write it as 0.416 .
The vinculum is written over the digit or digits that repeat.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 3 of 27
Example:
Convert
4
to a decimal.
11
This would be done by division.
0.3636...
4
 11 4.0000 or 0.36
11
Converting Decimals to Fractions
Syllabus Objective: (2.22) The student will write rational numbers in equivalent forms. (2.3)
The student will translate among different forms of rational numbers.
CCSS 8.NS.1-3: Convert a decimal expansion which repeats eventually into a rational
number.
To convert a decimal to a fraction:
1) Determine the denominator by counting the number of digits to the right of the
decimal point.
2) The numerator is the number to the right of the decimal point.
3) Simplify, if possible.
Examples:
1) Convert 0.52 to a fraction.
52
0.52 
100
=
13
25
2) Convert 0.613 to a fraction.
613
0.613 
1000
3) Convert 8.32 to a mixed number and improper fraction.
32
8.32  8
100
8
McDougal Littell, Chapter 5
8
208
or
25
25
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 4 of 27
But what if we have a repeating decimal?
While the decimals 0.3 and 0.3 look alike at first glance, they are different. They do not have the
3
same value. We know 0.3 is three tenths,
. How can we say or write 0.3 as a fraction?
10
As we often do in math, we take something we don’t recognize and make it look like a problem
we have done before. To do this, we eliminate the repeating part – the vinculum (line over the 3).
Example:
Convert 0.3 to a fraction.
0.3  0.333333...
Let’s let x = 0.333333...
Notice, and this is important, that only one number is repeating. If I
multiply both sides of the equation above by 10 (one zero), then subtract
the two equations, the repeating part disappears.
10 x  3.3333
 x  0.3333
9 x  3.0000
9 x 3.0000

9
9
1
x
3
1
is the equivalent fraction for 0.3
3
Example:
Convert 0.345 to a fraction.
The difficulty with this problem is the decimal is repeating. So we
eliminate the repeating part by letting x  0.345 .
0.345  0.345345345...
Note, three digits are repeating. By
multiplying both sides of the equation by
1000 (three zeros), the repeating parts line
up. When we subtract, the repeating part
disappears.
1000 x  345.345345345...

x
999 x 345

999 999
x
McDougal Littell, Chapter 5
0.345345345...
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
345 115
or
999 333
Page 5 of 27
Example:
Convert 0.13 to a fraction.
Note, one digit is repeating, but one is not. By multiplying both sides of
the equation by 10, the repeating parts line up. When we subtract, the
repeating part disappears.
10 x  1.33333
 x  .13333
9 x  1 .2
9 x 1.2

9
9
x
1.2
12 
2
or
 which simplifies to 
9
90 
15 
Ready for a “short cut”? Let’s look at some patterns for repeating decimals.
1
 0.111
9
or 01
.
1
 0.0909
11
or 0.09
2
 0.222
9
or 0.2
2
 0.1818
11
or 018
.
3
 0.333
9
or
3
 0.2727
11
or ?
4
 ?
9
?
4
 ?
11
It is easy to generate the missing decimals when you see the pattern!
Let’s continue to look at a few more repeating decimals, converting back into fractional form.
Because we are concentrating on the pattern, we will choose NOT to simplify fractions
where applicable. This would be a step to add later.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 6 of 27
0.5 
5
9
013
. 
13
99
0123
.

123
999
0.6 
6
9
0.25 
25
99
0154
.

154
999
0.7 
?
9
0.37 
?
99
0.421 
?
999
0.8 
?
?
0.56 
?
?
0.563 
?
?
The numerator of the fraction is the same numeral as the numeral under the vinculum. We can
also quickly determine the denominator: it is 9ths for one place under the vinculum, 99ths for two
places under the vinculum, 999ths for three places under the vinculum, and so on.
But what if the decimal is of a form where not all the numerals are under the vinculum? Let’s
look at a few.
0.23 
21
90
0.35 
32
90
0.427 
0.325 
423
990
322
990
0.4276 
0.235 
4272
9990
The numerator is generated by
subtracting the number not under the
vinculum from the entire number
(including the digits under the
vinculum).
We still determine the number of
nines in the denominator by looking
at the number of digits under the
vinculum. The number of digits not
under the vinculum gives us the
number of zeroes.
212
900
0.3759 
3722
9900
0.4276 
4234
9900
0.2015 
1814
9000
0.6024 
5964
9900
0.81437 
80623
99000
0.55341 
49807
90000
Note that again we chose not to simplify fractions where applicable as we want to
concentrate on the pattern.
Does
????
Do you believe it? Let's look at some reasons why it's true. Using the method we just looked at:
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 7 of 27
Surely if 9x = 9, then x = 1. But since x also equals .9999999... we get that .9999999... = 1.
But this is unconvincing to many people. So here's another argument. Most people who have
trouble with this fact oddly don't have trouble with the fact that 1/3 = .3333333... . Well,
consider the following addition of equations then:
This seems simplistic, but it's very, very convincing, isn't it? Or try it with some other
denominator:
Which works out very nicely. Or even:
It will work for any two fractions that have a repeating decimal representation and that add up to
1. The problem, though, is BELIEVING it is true.
So, you might think of 0.9999.... as another name for 1, just as 0.333... is another name for 1/3.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 8 of 27
Comparing and Ordering Rational Numbers
Syllabus Objectives: (2.24) The student will explain the relationship among equivalent
representations of rational numbers.
We will now have fractions, mixed numbers and decimals in ordering problems. Sometimes you can
simply think of (or draw) a number line and place the numbers on the line. Numbers increase as you
go from left to right on the number line, so this is particularly helpful when you are asked to go from
least to greatest.
If placement is not obvious (for instance, when values are very close together), it may be
advantageous to write all the number in the same form (decimal or fractional equivalents), and then
compare.
5
5 13
Example: Order the numbers  ,  0.2 , 4.3,  3, , 
from least to greatest.
4
2
3
Let’s first rewrite all improper fractions as mixed numbers.
5
1 5
1
13
1
  1 ;
 2 ;   4
4
4 2
2
3
3
Now let’s place the values on the number line.
4
1
3
3
1
5
1
0.2
4
2
1
2
0
From least to greatest, the order would be 
4.3
5
13
5
1
,  3,  ,  0.2 , 2 , 4.3 .
3
4
2
Sometimes writing the numbers in the same form will assist you in ordering.
Example: Order
7
5
, 0.25, 1,
, 1.1 from least to greatest.
8
11
(1) Find the decimal equivalents,
then compare.
7
 0.875
8
0.25  0.250
1  1.000
McDougal Littell, Chapter 5
1
 0.500
2
1.1  1.100
(2) Line up the decimals, the order from
least to greatest is:
0.250 0.500 0.875 1.000 1.100
(3) Use the original forms:
1 7
0.25, , , 1, 1.1
2 8
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 9 of 27
OR find the fractional equivalents and then compare.
7 35

8 40
0.25 
1
1 20

2 40
25 1 10
 
100 4 40
1.1 
11 44

10 40
40
40
Having found a common denominator, the order from least to greatest is:
10 20 35 40 44
,
,
,
,
40 40 Adding
40 40 and
40 Subtracting Fractions
with Like Denominators
Using the original forms:
1 7
0.25, , , 1, 1.1
2 8
Syllabus Objective: (2.4) The student will add fractions and mixed numbers. (2.5) The student
will subtract fractions and mixed numbers.
1
1
2
2
to . Will it be ? Why not? If we did, the fraction
would indicate that we
4
4
8
8
have two equal size pieces and that 8 of these equal size pieces made one whole unit. That’s not
true.
Let’s add
Let’s draw a picture to represent this:
1
4

1
4
Notice the pieces are the same size. That will allow us to add the pieces together. Each rectangle
has 4 equally sized pieces. Mathematically, we say that 4 is the common denominator. Now let’s
count the number of shaded pieces.
Adding the numerators, a total of 2 equally sized pieces are shaded and 4 pieces make one unit.
We can now show:
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 10 of 27
1
4

1
4
2 1
or
4 2
4 1
 .
9 9
Since the fractions have the same denominator, we write the sum over 9.
4 1 5
 
9 9 9
4
1
and .
Example: Find the difference of
5
5
Since the fractions have the same denominator, we write the difference over 5.
4 1 3
 
5 5 5
Example: Find the sum of
Writing these problems with variables does not change the strategy.
Example: Simplify the variable expression.
5x 2 x

12 12
5x 2 x 5x  2 x 7 x



12 12
12
12
Adding and Subtracting Fractions
with Unlike Denominators
Syllabus Objective: (2.4) The student will add fractions and mixed numbers. (2.5) The student
will subtract fractions and mixed numbers.
Let’s first review the ways to find a common denominator. We find the least common
denominator by determining the least common multiple.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 11 of 27
Strategy 1: Multiply the numbers. This is a quick, easy method to use when the numbers
are relatively prime (have no factors in common).
Example: Find the LCM of 4 and 5.
Since 4 and 5 are relatively prime, LCM would be 4  5 or 20 .
Strategy 2: List the multiples. Write multiples of each number until there is a common
multiple.
Example: Find the LCM of 12 and 16.
12, 24, 36, 48, 60, …
16, 32, 48, 64, …
48 is the smallest multiple of both numbers; therefore, 48 is the LCM.
Strategy 3: Prime factorization. Write the prime factorization of both numbers. The
LCM must contain all the factors of both numbers. Write all prime factors,
using the highest exponent.
Example: Find the LCM of 60 and 72.
60  22  3  5 and 72  23  32
The LCM is 23  32  5  360
This strategy can also be shown by using a Venn diagram.
Example: Find the LCM of 36 and 45.
Draw a Venn diagram, placing common factors in the intersection. The
LCM is the product of all the factors in the diagram.
Factors of 45
Factors of 36
22
32
5
Multiply all factors in diagram for the LCM: 22  32  5  180 .
As the numbers in the denominator become larger, this strategy can become
cumbersome. That is when the value of the following strategy becomes evident.
Strategy 4: Simplifying/Reducing Method. Write the two numbers as a single fraction;
then reduce and find the cross products. The product is the LCM.
Example: Find the LCM of 18 and 24.
18 3
 ;cross products are 18  4  24  3 or 72. The LCM is 72.
24 4
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 12 of 27
When adding or subtracting fractions, LCM is referred to as the Least Common Denominator
(LCD). We have several ways to find a common denominator.
Methods of Finding a Common Denominator
1. Multiply the denominators.
2. List multiples of each denominator, use a
common multiple.
3. Find the prime factorization of the
denominators, and find the Least Common
Multiple.
4. Use the Simplifying/Reducing Method.
Use this method when…
1. the denominators are prime numbers or
relatively prime.
2. the denominators are small numbers.
3. the denominators are small numbers;
some will advise to never or seldom use
this method.
4. the denominators are composite
numbers/ large numbers.
1
1
2
2
to . Will it be ? Why not? If we did, the fraction
would indicate that we
3
4
7
7
have two equal size pieces and that 7 of these equal size pieces made one whole unit. That’s just
not true.
Let’s add
Let’s draw a picture to represent this:
1
4

1
3
Notice the pieces are not the same size.
Making the same cuts in each rectangle will result in equally sized pieces. That will allow us to
add the pieces together. Each rectangle now has 12 equally sized pieces. Mathematically, we say
that 12 is the common denominator. Now let’s count the number of shaded pieces.
1 3

4 12

1 4

3 12
7
12
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 13 of 27
1
1
4
3
is the same as
and
has the same value as
.
3
4
12
12
Adding the numerators, a total of 7 equally sized pieces are shaded and 12 pieces make one unit.
From the drawing we can see that
If we did a number of these problems, we
would be able to find a way of adding and
subtracting fractions without drawing the
picture.
Algorithm for Adding/Subtracting
Fractions
1.
2.
3.
4.
Find a common denominator.
Make equivalent fractions.
Add/Subtract the numerators.
Simplify (reduce), if possible.
Using the algorithm, let’s try one.
Example:
1
5
2

3
Multiply the denominators to find the least common denominator, 5  3  15 . Now make
equivalent fractions and add the numerators.
1 3

5 15
2 10
 
3 15
13
15
These problems can also be written horizontally.
1 2 3 10 13
    .
5 3 15 15 15
Let’s try a few. Using the algorithm, first find the common denominator, and then make equal
fractions. Once you complete that, you add the numerators and place that result over the common
denominator and simplify, if possible.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 14 of 27
Remember, the reason you are finding a common denominator is so you have equally sized
pieces. To find a common denominator, use one of the strategies shown. Since the
denominators are relatively prime, use the “multiply the denominators” method.
Example:
3 15

4 20
1 4
 
5 20
19
20
3
4
1

5
Example: 11
14
5

8
To find the common denominator, use the Simplifying/Reducing Method,
8 4
 ; LCD  8  7  56
14 7
11 44

14 56
5 35
 
8 56
9
56
Writing these problems with variables does not change the strategy.
d 2d

3 5
The LCD is 15. Making equivalent fractions, we have:
Example: Simplify the expression.
d 5d

3 15

2 d 6d

5
15

McDougal Littell, Chapter 5
11d
15
It is customary
to write these
problems in a
horizontal format
like this →
d 2d 5d 6d



3 5
15 15
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
=
11d
15
Page 15 of 27
If the denominators are larger composite numbers, using the reducing method to find the
common denominator may make the work easier.
Example: Simplify the expression. 
5c 7c

18 24
Using the Simplifying/Reducing method:


5c
20c

18
72
18 3
 , 4 18  72 , so the LCD is 72.
24 4

or
5c 7c
20c 21c



18 24
72
72
7c
21c

24
72

=
41c
72
41c
72
Another nice feature of using the Simplifying/Reducing Method is that you do not need to
compute what 18   72 or 24   72 because we can see the number in the cross products.
That is, we can identify 18 times 4 is 72, so we multiply −5c by 4 to obtain the new
numerator ( 5c  4  20c ). Likewise, since 24 times 3 is 72, we determine the other
numerator as 7c  3  21c .
Example: Evaluate the expression.


3x
18 x

5
30
or
7x
21x

10
30
2x
4x


15
30

McDougal Littell, Chapter 5

3x 7 x 2 x


5 10 15

3x 7 x 2 x



5 10 15

18 x 21x 4 x
35 x
7x



or 
30
30 30
30
6
35 x
7x
or 
30
6
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 16 of 27
Regrouping To Subtract Mixed Numbers
Syllabus Objective: (2.5) The student will subtract fractions and mixed numbers.
The concept of borrowing when subtracting with fractions has been typically a difficult area for
kids to master. For example, when subtracting 12 1 6  4 5 6 , students usually answer 8 4 6 if they
subtract this problem incorrectly. In order to ease the borrowing concept for fraction, it would be
a good idea to go back and review borrowing concepts that kids are familiar with.
Example: Take away 3 hours 47 minutes from 5 hours 16 minutes.
5 hrs 16 min
 3 hrs 47 min
?????????
Subtracting the hours is not a problem but students will see that 47 minutes cannot
be subtracted from 16 minutes. In this case, students will see that 1 hour must be
borrowed from 5 hrs and added to 16 minutes:
4hrs
5 hrs 16 min16min 1hr  16min 60min  76min
 3 hrs 47 min
?????????
Now the subtraction problem can be rewritten as:
4 hrs 76 min
4 hrs 76 min
 3 hrs 47 min
 3 hrs 47 min
???????????
1 hr 29 min
If students can understand the borrowing concept from the previous example, the same concept
can be linked to borrowing with mixed numbers. Lets go back to the first example: 12 1 6  4 5 6 .
It may be easier to link the borrowing concept if the problem
1 1
1 6 7
11 12
1   
7
6 6 6
6 6
11
is rewritten vertically:
6
5
 4
5
6
 4
6
???????
7
McDougal Littell, Chapter 5
2
1
7
6
3
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 17 of 27
Example: Subtract
2
5
1
7
2
13
Step 1. Find a common denominator:
The common denominator is 10.
Step 2. Make Equivalent fractions using 10 as
the denominator.
4
10
5
7
10
13
Step 3. It is not possible to subtract the
numerators. You cannot take 5 from 4!!
Use the concept of borrowing as
described in the above examples to rewrite this problem. Borrow from 1
10
4
from 13 and add 1 ( ) to
.
10
10
4 10

10 10
5
7
10
12 13
14
10
5
7
10
12
5
9
10
1
3
cups of flour. She used 1 cups of flour to
2
4
bake a cake. How much flour is left in the canister?
Example: Catherine has a canister filled with 5
Subtract 5
1
3
1 .
2
4
Step 1. Find a common denominator: The common denominator between 2 and 4 is 4.
Step 2. Make equivalent fractions using 4 as the common denominator.
2
3
5 1
4
4
Step 3. When subtracting the numerators, it is not possible to take 3 from 2, therefore
borrow. It may be easier to follow the borrowing if the problem is rewritten
vertically .
2 4
4 5 4
4
3
1
4
6
4
3
1
4
4
3
There are 3
3
4
3
cups of flour left in the canister.
4
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 18 of 27
Multiplying Fractions and Mixed Numbers
Syllabus Objective: (2.6) The student will multiply fractions and mixed numbers.
Multiplying fractions is pretty straight-forward. So, we’ll just write the algorithm for it, give an
example and move on.
Algorithm for Multiplying Fractions and Mixed
Numbers
1.
2.
3.
4.
5.
Example: 3
1
2
Make sure you have proper or improper fractions.
Cancel, if possible.
Multiply numerators.
Multiply denominators.
Simplify (reduce), if possible.
4
5
Since 3
7
2
1
7
is not a fraction, we convert it to .
2
2
4
can be written as
5
3
1
2
4 7

5 2
4
5
7 4
2 5
Now what I’m about to say is important and will make your life a lot easier. We know
how to reduce fractions, so what we want to do now is cancel with fractions. That’s
nothing more than reducing using the commutative and associative properties.
Using the commutative property, we can rewrite this as
Using the associative property, we can rewrite this as
Simplify
4 7
.
2 5
4 7
.
2 5
4 2
 .
2 1
Then multiply and simplify, as a mixed number.
2
1
7 14
4

2
5 5
5
Rather than going through all those steps, we could take a shortcut and cancel.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 19 of 27
Now rather than going through all those steps, using the commutative and associative properties,
we could have taken a shortcut and cancelled.
2
7
4

5
2
1
To cancel, we would look for common factors in the numerator and the denominator and divide
them out. In our problem, there is a common factor of 2. By dividing out a 2, the problem looks
like this:
7
1
2 14
4

or 2
5 5
5
Let’s look at another one.
Example:
3
3
2
2
5
9
18
5
2
1
18
5
2
1
20
9
Rewrite as improper fractions.
4
20
9
1
4 8
 8
1 1
Cancel 18 and 9 by common factor of 9.
Cancel 20 and 5 by common factor of 5.
Multiply numerators, multiply denominators, simplify.
When variables are added to these problems, the strategy remains the same.
Example: Simplify the expression.
1
3n 2 2 n 2 3n 4


4
7
14
2
McDougal Littell, Chapter 5
3n 2 2n 2

4
7
Cancel the 2 and 4 by common factor of 2.
Multiply the numerators and denominators.
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 20 of 27
Dividing Fractions and Mixed Numbers
Syllabus Objective: (2.7) The student will divide fractions and mixed numbers.
Before we learn how to divide fractions, let’s revisit the concept of division using whole
numbers. When I ask, how many 2’s are there in 8, I can write that mathematically three ways.
28
8
2
82
To find out how many 2’s there are in 8, we will use the subtraction model:
8
Now, how many times did we subtract 2? Count them: there are 4 subtractions.
2
So there are 4 twos in eight.
6
Mathematically, we say 8 ÷ 2 = 4.
2
4
2
2
The good news is, division has been defined as repeated subtraction That won’t
change because we are using a different number set. In other words, to divide
fractions, I could also do repeated subtraction.
2
0
1 1
Example: 1 
2 4
Another way to look at this problem is using your experiences with money. How
many quarters are there in $1.50? Using repeated subtraction we have:
4
4
1

4
3
4
1

4
2
4
1
2
1 1
2
4
1

4
1
1
4
1

4
1
1
How many times did we subtract
2
4
1

4
1
4
1

4
0
1
1 1
? Six. Therefore, 1   6 . But this took
4
2 4
a lot of time and space.
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 21 of 27
A visual representation of division of fractions would look like the following.
Example:
1 1
 
2 8
We have
1
. Representing that would be
2
1
1
' s are there in , we
2
8
need to cut this entire diagram into eighths. Then count each of the shaded
one-eighths.
Since the question we need to answer is how many
As you can see there are four. So
1 1
  4.
2 8
5 1
 
6 3
Example:
We have
5
. Representing that would be
6
Since the question we need to answer is how many
5
1
' s are there in , we need to use the cuts
6
3
for thirds only. Then count each of the one-thirds.
1
2
1
2
1
5 1
1
As you can see that are 2 . So   2 .
2
6 3
2
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 22 of 27
Be careful to choose division examples that are easy to represent in visual form.
Well, because some enjoy playing with numbers, they found a quick way of dividing fractions.
They did this by looking at fractions that were to be divided and they noticed a pattern. And here
is what they noticed.
Algorithm for Dividing Fractions and Mixed
Numbers
1. Make sure you have proper or improper
fractions.
2. Invert the divisor (2nd number).
3. Cancel, if possible.
4. Multiply numerators.
5. Multiply denominators.
6. Simplify (reduce), if possible.
The very simple reason we tip the divisor upside-down, then multiply for division of fractions is
because it works. And it works faster than if we did repeated subtractions, not to mention it
takes less time and less space.
Example:
3 2
3 5

4 2
4 5
(Invert the divisor.)
1 4
Example: 3 
3 9
Multiply numerators and
denominators, and simplify.
10 4
  Make sure you have proper or improper fractions.
3 9
5
1
10
3
9
 Invert the divisor.
4
10
3
93
 Cancel 10 and 4 by 2, and cancel 9 and 3 by 3.
4
2
5
1
McDougal Littell, Chapter 5
15
7
1
8
8
3

2
15

2
1
7
2
Multiply numerators and denominators.
Simplify.
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 23 of 27
Computing with Fractions and Signed Numbers
Syllabus Objective: (2.6) The student will multiply fractions and mixed numbers. (2.7) The
student will divide fractions and mixed numbers.
The rules for adding, subtracting, multiplying and dividing fractions with signed numbers are the
same as before, the only difference is you integrate the rules for integers.
3 2
3
Example:  

4 7 Invert divisor 4
9  25 
Example: 
 
10  12 
7
2
5
3
9  25 

 
10  12 
2
4
21
5
 2
8
8

Multiply numerators and
denominators, and
simplify.
3  5
15
7
    1
2  4
8
8
Solving Equations and Inequalities
Containing Fractions and Decimals
Syllabus Objective: (2.8) The student will use the multiplicative inverse to solve equations with
fractional coefficients. (2.9) The student will solve equations and inequalities with rational
numbers.
First Strategy for Solving: You solve equations and inequalities containing fractions and
decimals the same as you do with whole numbers; the strategy does not change. To solve linear
equations or inequalities, put the variable terms on one side of the equal sign, and put the
constant (number) terms on the other side. To do this, use opposite (inverse) operations.
Example: Solve: x 
1
2
 .
3
5
1
2
6
 
3
5
15
1
1
5

 
3
3
15
11
x
15
x
McDougal Littell, Chapter 5
Undo adding one-third by subtracting onethird from both sides of the equation; make
equivalent fractions with a common
denominator of 15.
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 24 of 27
Example: Solve:
x 2

5 3
x

5
 5   x 
 1   5  



2
3
2  5 
 
3 1 
10
1
x
or  3
3
3
Example: Solve:
Undo dividing by –5 by multiplying both
sides by –5. Cancel.
Multiply numerators and denominators,
and simplify.
2
x  4  20
3
2
x  4  20
3
4 4
2
x  24
3
 3
2
x   3 24
3
2 x  72
2 x 72

2
2
x  36
We could have saved a little time by recognizing that multiplying by 3 and then dividing by 2
3
could have been done in one step by multiplying by the reciprocal .
2
2
x  4  20
3
4 4
2
x  24
3
 32
 3
  x    24
23
2
x  36
McDougal Littell, Chapter 5
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 25 of 27
Example: Solve 2.5  5.3  0.7 x
2.5  5.3  0.7 x
 5.3   5.3
 2 . 8  0. 7 x
 2 . 8 0 .7 x

0.7
0.7
4  x
Second Strategy for Solving: Another way to solve an equation or inequality with fractions is
to “clear the fractions” by multiplying both sides of the equation or inequality by the LCD of the
fractions. The resulting equation/inequality is equivalent to the original. You can also clear
decimals by determining the greatest number of decimal places and multiplying both sides of the
equation/inequality by that power of 10.
Example: Solve
2
5
x5 .
3
2
2
5
x5
3
2
Original equation.
2

5
6  x  5  6  
3

2
2 
5
6  x   6  5  6  
3 
2
Multiply each side by LCD of 6.
Distribute.
Simplify.
4 x  30  15
30  30
Undo adding 30 by subtracting 30from both sides.
4 x  15
4 x 15

4
4
x  3
McDougal Littell, Chapter 5
3
4
Undo multiplying by 4 by dividing by 4.
Simplify.
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 26 of 27
Example: Solve 5.14  0.8x  2.3 .
5.14  0.8 x  2.3
Original inequality.
100  5.14  0.8 x   100  2.3
514  80 x  230
514
Since greatest number of decimals is 2, multiply by 102 or 100.
Distribute and simplify.
 514
Undo addition by subtracting 514 from each side.
80 x  284
Simplify.
80 x 284

80
80
Undo multiplication by dividing both sides by 80.
x  3.55
Simplify.
Example: Solve 2.875  9  12.45 .
2.875 x  9  12.45
Original inequality.
1000  2.875 x  9   1000 12.45 Since greatest number of decimals is 3, multiply by 103 or 1000.
2875 x  9000  12450
9000  9000
Distribute and simplify.
Undo addition by subtracting 9000 from each side.
2875 x  3450
Simplify.
2875 x 3450

2875 2875
Undo multiplication by dividing both sides by −2875. Reverse
the inequality.
x  1.2
McDougal Littell, Chapter 5
Simplify.
Pre-Algebra 8, Unit 05: Rational Numbers and Equations
Revised 2012 - CCSS
Page 27 of 27