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Transcript
Math 7 Notes – Unit Three: Applying Rational Numbers
Strategy note to teachers: Typically students need more practice doing computations with
fractions. You may want to consider teaching the sections on fractions first, and then introduce
decimals as special fractions (rational numbers equivalent to fractions). This would allow you to
reinforce the rules for fractions while teaching the sections on decimals. For instance, you could
define a decimal as a special fraction and link all the rules introduced in fractions to decimals.
That is when you line up decimal points and fill in zeros when adding or subtracting decimals,
that’s the same as finding a common denominator and making equivalent fractions when adding
or subtracting fractions, etc.
Estimating with Decimals
Syllabus Objective: (1.8) The student will estimate using a variety of methods.
An estimation strategy for adding and subtracting decimals is to round each number to the
nearest integer and then perform the operation.
Examples: 25.8

26
 5.2

5
 8.98   9
14.2

14
 3.7

4
9.2
1
12
1 would be our estimate
12 would be our estimate

9
 18
-18 would be our estimate
An estimation strategy for multiplying and dividing decimals:
 When multiplying, round numbers to the nearest non-zero number or to numbers that are
easy to multiply.
 When dividing, round to numbers that divide evenly, leaving no remainders.
 Remember, the goal of estimating is to create a problem that can easily be done
mentally.
Examples:
38.2

40
 6.7

7
280
280 would be our estimate
Math 7, Unit 03: Applying Rational Numbers
 33.6  4.2

 32  4  8
8 would be our estimate
Holt: Chapter 3
Page 1 of 24
Estimating can be used as a test-taking strategy. Use estimating to calculate your answer first, so
you can eliminate any obviously wrong answers.
Example: A shopper buys 3 items weighing 4.1 ounces, 7.89 ounces and 3.125 ounces.
What is the total weight?
A. 0.00395
B. 3.995
C. 14.0
D. 15.115
Round the individual values:
4.1  4
7.89  8
3.125  3
15
The only two reasonable answers are C) and D). Eliminating unreasonable answers can
help students to avoid making careless errors.
Adding and Subtracting Decimals
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Remember, since decimals are special fractions, the rules for computing with fractions also work
for decimals.
Note: Choose numbers carefully that are not reducible as fractions, so as to eliminate that
distraction.
Example: To add .71 and .127, follow the algorithm for adding fractions:
Algorithm for Addition of Fractions
1.
2.
3.
4.
5.
Find the common denominator.
Make equivalent fractions.
Add the numerators.
Bring down the denominator.
Simplify.
Math 7, Unit 03: Applying Rational Numbers
71
710

100
1,000
127
127


1,000
1,000
.71
.127
.710
 .127
.837
837
1,000
Holt: Chapter 3
Page 2 of 24
1. The denominator for .71 is 100. The denominator for .127 is 1,000.
The least common denominator is 1,000.
2. Add a zero after .71 so it will have a common denominator of 1,000.
3. Add the numerators by adding 710 and 127 = 837.
4. Bring down the common denominator of 1,000. This means there must be three digits to
the right of the decimal point. Placing a decimal point gives .837 as the answer.
After doing numerous decimal addition problems we can see a pattern which leads to a simpler
algorithm that can be used to add/subtract decimals.
Algorithm for Addition / Subtraction of Decimals
1. Rewrite the problems vertically, lining up the decimal
points.
2. Fill in spaces with zeros.
3. Add or subtract the numbers.
4. Bring the decimal point straight down.
***When comparing the algorithm for adding/subtracting fractions and the algorithm for
adding/subtracting decimals, we note that:
1. Lining up the decimal point is the same as finding the common denominator in fractions.
2. Filling in spaces with zeros is the same as making equivalent fractions.
3. Adding or subtracting the numbers is the same as adding the numerators.
4. Bringing the decimal point straight down is the same as bringing down the denominator.
Example: 1.23  .4  12.375
Rewrite vertically, lining up the decimal points.
1.23
.4
12.375
1. Fill in with zero to find the common denominator and make
equivalent fractions.
1.230
.400
12.375
2. Add and bring the decimal straight down.
1.230
.400
12.375
14.005
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 3 of 24
Example: 15  1.5
15.
15.0
15.0
1.5
1.5
1.5
13.5
Multiplying Decimals
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Just as the algorithm for adding and subtracting decimals is related to the addition and
subtraction of fractions, the algorithm for the multiplication of decimals also comes directly from
the multiplication algorithm for fractions. Before we look at the fraction algorithm, let’s first
look at the algorithm for decimal multiplication.
Algorithm for Multiplication of Decimals
1. Rewrite the numbers vertically.
2. Multiply normally, ignoring the decimal point.
3. Count the total number of digits to the right of the
decimal points in the factors.
4. Count and place the decimal point that same number of
places from right to left in the product (answer).
4.2 1.63
Example:
1.63
1. Rewrite the problem vertically.
 4.2
2. Multiply normally, ignoring the decimal point.
326
652
6846
6. 8 4 6
3. Count the number of digits to the right of the decimal points.
There are two places to the right in the multiplicand (number
on the top) and one place to the right in the multiplier (number
on the bottom). 2  1  3
4. Count and place the decimal point the same number of places
(3) from right to left in the answer.
Before going on, let’s think of how this is related to the algorithm for multiplying fractions. The
algorithm for multiplying fractions is to multiply the numerators, multiply the denominators, and
then simplify. Multiplying the numbers while ignoring the decimal points is the same as
multiplying the numerators. Counting the number of decimal places is the same as multiplying
the denominators. In this problem, the denominators are 100 and 10. 100 10  1000 which
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 4 of 24
shows three zeros, which means 3 decimal places. The point is the algorithm for multiplying
decimals comes from the algorithm from multiplication of fractions. That should almost be
expected since decimals are special fractions.
Remember to point out to students that if a number has no decimal point, the decimal point is
understood to go after the number.
Example: 15  15. So, when multiplying this number, it has NO decimal places after the
decimal point.
Multiplying by Positive Powers of 10
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Examining the patterns discovered when multiplying by positive powers of 10 (10, 100, 1,000 …)
we can do many multiplication problems mentally. What pattern do you see from these problems?
10 (12.34) = 123.4
10 (0.056) = 0.56
100 (567.234) = 56723.4
100 (54.7) = 5470
1,000 (4.56) = 4560
1,000 (0.3456) = 345.6
We can generalize a rule from the observed pattern: When multiplying by positive powers of 10,
move the decimal point to the right, the same number of places as there are zeros.
Example: 10 (123.75)
One zero in 10, so move the decimal point one place to the
right.
10 (123.75) = 1237.5
Example: 100 (5.237)
Two zeros in 100, move the decimal point two places to the
right.
100 (5.237) = 523.7
Example: 1000 (16.2)
1000 (16.2) = 16,200
Math 7, Unit 03: Applying Rational Numbers
Three zeros in 1,000; move the decimal point three places
to the right.
Notice in this problem we had to fill in a couple of
placeholders to move it three places to the right.
Holt: Chapter 3
Page 5 of 24
Dividing Decimals by Integers
Syllabus Objective: (2.1)The student will solve problems using operations on positive and
negative numbers, including rationals.
Algorithm for Dividing Decimals by Integers
1. Bring the decimal point straight up into the quotient.
2. Divide in the normal way and determine the sign.
Example: 7 3.92
Or use short division:
.56
7 3.92
35
42
42
Math 7, Unit 03: Applying Rational Numbers
.5
7 3.92
.5 6
7 3.942
1) Place decimal point in quotient.
2) 7 divides into 39, 5 times with 4 left.
3) Write remainder of 4, before 2.
4) 7 divides into 42, 6 times evenly.
Remember, when using short division,
writing the remainder(s) in the dividend
is showing work.
Holt: Chapter 3
Page 6 of 24
Dividing Decimals and Integers by Decimals
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Algorithm for Dividing Decimals
1. In the divisor, move the decimal point as far to the right as possible.
2. In the dividend, move the decimal point the same number places to the
right.
3. Bring the decimal point straight up into the quotient.
4. Divide in the normal way.
Example:
.12 .456
12 .456
Move the decimal point 2 places to the right in the divisor.
Move the decimal point 2 places to the right in the dividend.
12 45.6
Bring up the decimal point and divide normally.
3.8
12 45.6
36
96
96
By moving the decimal the same number of places to the right in the divisor
and the dividend, we are essentially multiplying our original expression by
one. We are making equivalent fractions by multiplying the numerator and
denominator by the same number. If we move the decimal point one place,
we are multiplying the numerator and denominator by 10. By moving it
two places, we are multiplying the numerator and denominator by 100, etc.
.456 100 45.6


.12 100 12
Dividing by Powers of 10
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Examining the patterns discovered when dividing by powers of 10 (10, 100, 1,000 …) we can do
many division problems mentally. What pattern do you see from these problems?
67.89 ÷ 10 = 6.789
654 ÷ 10 = 65.4
2398.6 ÷100 = 23.986
78 ÷ 100 = .78
Math 7, Unit 03: Applying Rational Numbers
2468 ÷ 1000 = 2.468
8 ÷ 1000 = .008
Holt: Chapter 3
Page 7 of 24
We can generalize a rule from the above observed pattern: when dividing by positive powers of
10, move the decimal point to the left, the same number of places as there are zeros.
Example: 5.6 ÷ 10
Since there is one zero in 10, move the decimal point one place to the left.
5.6 ÷ 10 = .56
Example: 9832 ÷ 100
Since there are two zeros in 100, move the decimal point two places to the left.
9832 ÷ 100 = 98.32
Example: 92 ÷ 1000
There are three zeros in 1,000, so move the decimal point three places to the left.
9.2 ÷ 1000 = .0092
Notice in this problem we had to fill in a couple of
placeholders to move it three places to the left.
Computing with Decimals and Signed Numbers
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
The rules for adding, subtracting, multiplying and dividing decimals with signed numbers are the
same as before, the only difference is you integrate the rules for integers.
Example:
Compute: 6.7  ( 41.23)
–6.70
+ – 41.23
– 47.93
Example:
Line up decimal points, fill in zeros, and add.
Add 2 negative numbers, the sum is negative.
Compute: 9.376  ( 7.2)
 9.376

7.200
 2.176
Math 7, Unit 03: Applying Rational Numbers
Since subtract means add the opposite, we add 7.2
Line up decimal points, fill in zeros, and add.
Adding numbers with different signs, we subtract
the absolute values and take the sign of the greater
absolute value so our sum is negative.
Holt: Chapter 3
Page 8 of 24
Compute: ( 5.6)( 8)
Example:
 5.6

Multiply, count one decimal place in problem and place one
decimal place in answer.
8
 44.8
Example:
Multiply 2 negatives, the product is positive.
Compute: 7.5  .15 
50.
.15 7.5  15 750.
 75
00
 00
0
Move decimal point two places to the right in the
divisor to make it a whole number.
Move decimal point two places to the right in the
dividend.
Bring decimal point straight up into the quotient.
Divide as usual
Determine the sign.
Solving Equations Containing Decimals
Syllabus Objective: (3.5) The student will solve equations and inequalities in one variable
with integer solutions.
Strategy for Solving Equations: To solve linear equations, 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:
Example:
by 8 .
x + 1.5 = –7.34
–1.5 –1.50
x = –8.84
.48  8n
.48 8n

8 8
0.06  n
Math 7, Unit 03: Applying Rational Numbers
Undo adding 1.5 by subtracting 1.5 from both sides.
Simplify and determine the sign.
Undo multiplying by 8 by dividing both sides
Simplify and determine the sign.
Holt: Chapter 3
Page 9 of 24
Example:
y
 0.7
5
y
( 5)
 ( 5)( 0.7) Undo dividing by –5, by multiplying both sides by –5.
5
Simplify and determine the sign.
y  3.5
Estimation with Fractions
Syllabus Objective: (1.8) The student will estimate using a variety of methods.
Benchmarks for Rounding Fractions
Round to 0 if the
Round to ½ if the
Round to 1 if the
numerator is much smaller numerator is about half the numerator is nearly equal
than the denominator.
denominator.
to the denominator.
3 1 7
11 5 47
17 7 87
Examples:
Examples:
Examples:
, ,
, ,
, ,
20 8 100
20 9 100
20 8 100
Examples:
Round to 0, ½ or 1
1.
7
1
9
2.
27 1

50 2
3.
2
0
25
Estimating Sums and Differences
Round each fraction or mixed number to the nearest half, and then simplify using the rules for
signed numbers.
Example: Estimate
7 1
 .
8 9
Math 7, Unit 03: Applying Rational Numbers
7
1
8
1
 0
9
1 is our estimate
Holt: Chapter 3
Page 10 of 24
3
1
Example: Estimate 5  8
4
7
3
5  6
4
1
 8  8
7
 2 is our estimate
Estimating Products and Quotients
Round each mixed number to the nearest integer, and then simplify.
Example:
5
1
7  11  8  11
6
5
 88 is our estimate
Example:
5
4
17  ( 2 )  18  ( 3)
6
5
 6 is our estimate
Adding and Subtracting Fractions
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Common Denominators
Let’s say we have two cakes, one chocolate and the other vanilla. The chocolate cake was cut
into fourths, the vanilla cake into thirds as shown below. You take one piece of each, as shown.
Since you had 2 pieces of cake, can you say you had
2
of a cake?
7
Remember our definition of a fraction:
 the numerator indicates the number of equal size pieces you have
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 11 of 24

while the denominator indicates how many equal pieces make one whole cake
Since your pieces are not equal, we can’t say you have
2
of a cake.
7
And clearly 7 pieces does not make one whole cake. We can conclude:
1 1 2
  .
4 3 7
The key is to cut the cakes into equally sized pieces.
We’ll cut the first cake (already in fourths) the same way the second cake was cut. And we’ll cut
the second cake (already in thirds) the same way the first cake was cut. So each cake ends up
being cut into twelve equally sized pieces.
Cutting the cake into EQUAL size pieces illustrates the idea of common denominator.
Let’s look at several different methods of finding a common denominator.
Methods of Finding a Common Denominator
A common denominator
is a denominator that all
other denominators will
divide into evenly.
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
3. Use the Simplifying/Reducing Method, especially for larger
denominators.
In our cake illustration, the common denominator is the number of pieces tin which the cakes can
be cut so that everyone has the same size piece.
1
1
and , multiply the denominators, 3  4  12 .
3
4
This technique is very useful when the denominators do not share any common
factors other than 1.
Method 1: To find a common denominator of
Method 2: To find a common denominator of
5
3
and , list the multiples of each denominator.
6
4
Multiples of 6: 6, 12, 18, 24…
Mulitples of 4: 4, 8, 12, …
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 12 of 24
Since 12 is on each list of multiples, it is a common denominator.
This technique is very useful when the greater denominator is a multiple of the
smaller one.
3
5
Such as, find the common denominator for
and
. 16 is a common
4
16
denominator since it is a multiple of 4.
Method 3: To find a common denominator of
3
7
and
, find the prime factorization of each
8
12
denominator.
Factoring,
8  222
12  2  2  3
Multiply the prime factors, using overlapping
factors only once, we get 2  2  2  3  24 .
This technique can be tedious and may be the least desirable method; although it is good
practice working with prime factors.
1
5
and
using the Simplifying Method,
18
24
18
18 6 3
  .
create a fraction using the two denominators:
, and then simplify:
24 6 4
24
18 3
For
 , cross multiply: 4 18  72 or 3  24  72 . The common denominator is
24 4
72.
18
24
or
.
Note: It does not matter if you use
24
18
Method 4: To find the common denominator of
This is an especially good way of finding common denominators for fractions that
have large denominators or fractions whose denominators are not that familiar to
you.
Adding and Subtracting Fractions with Like Denominators
To add or subtract fractions, you must have equal pieces. If a cake is cut into 8 equal pieces and
you eat three pieces today, and then eat four pieces tomorrow, you would have eaten a total of 7
7
pieces of cake, or of the cake.
8
3 4 7
+ 
8 8 8
The numerators are added to indicate the number of equal pieces that were eaten.
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 13 of 24
The denominators are NOT added, because the denominator indicates the number of equal pieces
in the cake. If we added them, we would get 16, but there are only 8 pieces of cake.
Examples:
1 2 3
+ 
8 8 8
6
3
3


10 10 10
Adding and Subtracting Fractions with Unlike Denominators
1
1
2
to . Do we get ?
3
4
7
What does the picture tell us?
Let’s add
1
4
+
1
3
Note that when creating your visuals, be sure to cut one figure vertically and the other
horizontally.
The pieces are NOT EQUAL, so we cannot add them. We need to divide each shape into
equally sized pieces. Do this by using the horizontal cuts from the second visual on the first.
Then use the vertical cuts from the first visual on the second. (Both will now be divided into
twelfths.)
+
1
4
1 3

4 12
1
3
1 4
 
3 12
7
12
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 14 of 24
1
1
3
4
is the same as
and has the same value as
. Adding
4
3
12
12
the numerators, a total of 7 equally sized pieces are shaded and 12 pieces make one unit. If you
did a number of these problems, you would be able to find a way of adding and subtracting
fractions without drawing the picture.
From the picture we can see that
Algorithm for Adding / Subtracting Fractions
1.
2.
3.
4.
5.
Example:
Examples:
Find a common denominator.
Make equivalent fractions.
Add/subtract the numerators.
Bring down the denominator.
Simplify.
1
5
2

3
1

9
5
 
12
1. To find the common denominator, multiply the
denominators, since they are relatively prime (have no
common factors greater than 1). (3  5  15)
2. Make equivalent fractions. (shown)
3. Add the numerators. (3 + 10 = 13)
4. Bring down the denominator. (15)
5. Simplify. (not necessary in this problem)
4
36
15
36
19
36
Math 7, Unit 03: Applying Rational Numbers
1 3

5 15
2 10
 
3 15
13
15
7
21

24 72
5 10
 
36 72
11
72
Holt: Chapter 3
Page 15 of 24
Adding and Subtracting Mixed Numbers
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Algorithm for Adding/Subtracting Mixed
Numbers
1.
2.
3.
4.
5.
6.
Example:
Find a common denominator.
Make equivalent fractions.
Add/subtract the whole numbers.
Add/subtract the numerators.
Bring down the denominator.
Simplify the fraction answer part,
combining it with the whole number part.
To find the common
denominator, multiply the
denominators, since they are
relatively prime (have no
common factors greater than 1).
4•5 = 20
3
5
3
3
4
2
Example:
3
12
2 2
5
20
3
15
3  3
4
20
27
7
5 6
20
20
4
16
5
9
36
7
21
3  3
12
36
37
1
8
9
36
36
5
Borrowing From a Whole Number
Subtracting fractions and borrowing is as easy as getting change for your money.
Example: You have 8 one dollar bills and you have to give your friend $3.25. How
much money would you have left?
Since you don’t have any coins, you would have to change one of the dollars into 4 quarters.
Why not ten dimes? Because you have to give your friend a quarter, so you get the change in
terms of what you are working with – quarters.
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 16 of 24
8 dollars
– 3 dollars 1 quarter
7 dollars 4 quarters
– 3 dollars 1 quarter
Subtract to get: 4 dollars 3 quarters
Redoing this problem using fractions:
4
4
1
3
4
3
4
4
8
3
7
1
4
In the last problem, we borrowed
quarters because we were working
with quarters. Now we will borrow
4ths for the same reason.
Example:
5
5
2
9
5
3
2
5
12
11
2
 9
5
5
We change 12 to 11 because we are
5
working with fifths.
Borrowing from Mixed Numbers
Example: For this problem you have 6 dollars and 1 quarter in your pocket, and you have
to give your brother $2.75.
6 dollars 1 quarter
5 dollars 5 quarters
– 2 dollars 3 quarters
– 2 dollars 3 quarters
Subtract to get: 3 dollars 2 quarters
1
4
3
2
4
6
Math 7, Unit 03: Applying Rational Numbers
Since we don’t have enough quarters, we
change 1 dollar for 4 quarters, adding
that to the quarter we already had, that
gives you 5 quarters.
1
5
4
1 4
5
We change 6 to 5 and add the to make 5 .
 5
4
4
4
4
4
3
3
2
=2
4
4
2
1
3 3
when we simplify.
4
2
5
Holt: Chapter 3
Page 17 of 24
Algorithm for Borrowing with Mixed Numbers
1.
2.
3.
3.
4.
5.
6.
Find a common denominator.
Make equivalent fractions.
Borrow, if necessary.
Subtract the whole numbers.
Subtract the numerators.
Bring down the denominator.
Simplify if needed.
This example has unlike denominators.
Example:
3  12
15
5
12
12
8
8
2
=2
12
12
7
3
12
1
3
 6
4
12
2
8
2   2
3
12
5
6
Multiplying Fractions and Mixed Numbers
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
Before we learn how to multiply fractions, let’s revisit the concept of multiplication using whole
numbers. When I have an example like 3  2  we can model that in several ways.
Since it can be read three
groups of 2 we can use
the addition model.
We can show a rectangular array
2
2
2
Show 3 groups of 2
3
2
6
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 18 of 24
Each representation shows a total of 6.
Mathematically, we say 3  2  6 .
Multiplication is defined as repeated addition. That won’t change because we are using a
different number set. In other words, to multiply fractions, I could also do repeated addition.
Example: 6 
1
1 1 1 1 1 1
6

     
3
2
2 2 2 2 2 2
2
A visual representation of multiplication of fractions would look like the following.
Example:
1 1
 
2 3
In this example, we want half of one third. So the visual begins with the one third we have.
Since we want only half of it, we cut the visual in half and shade.
1
Now we see 1 part double shaded and 6 total parts or .
6
Example:
1 1
 
3 4
1
. So the visual begins with the one fourth.
4
We want one third of it.
In this example, we want to take one third of
Now we visually see 1 part doubly shaded, and 12 total pieces. So
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
1 1 1
  .
3 4 12
Page 19 of 24
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.
4
5
3
1
2
4 7

5 2
4
5
Since 3
it to
7
2
1
is not a fraction, we convert
2
7
.
2
4
7 4
can be written as
5
2 5
7 4
.
5 2
4 2
7 4
.
Using the associative property, we can rewrite this as
Simplify  .
5 2
2 1
2 7 14
4

2
Then multiply and simplify, as a mixed number.
1 5 5
5
Using the commutative property, we can rewrite this as
Rather than going through all those steps, we could take a shortcut and cancel.
Example: 3
3
2
2
5
9
18
5
2
1
2
1
18
5
20
9
20
9
Make sure you have proper or improper fractions.
4
Cancel 18 and 9 by common factor of 9.
1
4 8
 8
1 1
Math 7, Unit 03: Applying Rational Numbers
Cancel 20 and 5 by common factor of 5.
Multiply numerators, multiply denominators, simplify.
Holt: Chapter 3
Page 20 of 24
Dividing Fractions and Mixed Numbers
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
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. So
2
there are 4 twos in eight.
6
Mathematically, we say 8 ÷ 2 = 4.
2
4
2
2
Division is 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:
1 2
1 1
2 4
1

4
1
1
4
1

4
1
How many times did we subtract
Math 7, Unit 03: Applying Rational Numbers
4
4
1

4
3
4
1

4
2
4
1
2
4
1

4
1
4
1

4
0
1
? Six. But this took a lot of time and space.
4
Holt: Chapter 3
Page 21 of 24
A visual representation of division of fractions would look like the following.
Example:
We have
1 1
 
2 8
1
. Representing that would be
2
1
1
' s are there in , we need to cut this entire
2
8
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
Example:
We have
1 1
  4.
2 8
5 1
 
6 3
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
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 22 of 24
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.
2.
3.
4.
5.
6.
Make sure you have proper or improper fractions.
Invert the divisor (2nd number).
Cancel, if possible.
Multiply numerators.
Multiply denominators.
Determine your sign and simplify.
The very simple reason we tip the divisor upside-down (use the reciprocal), 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.
3 2
3 5

Example:  
4 5
4 2
(Invert the divisor.)
1 4
Example: 3 
3 9
5
1
5
1

15
7
 1
8
8
Multiply numerators and
denominators, and simplify.
10 4

3 9
Make sure you have proper or improper fractions.
10
3
9
4
Invert the divisor.
10
3
93
Cancel 10 and 4 by 2, and cancel 9 and 3 by 3.
4
2
3 15

2 2
Multiply numerators and denominators.
15
1
7
Simplify.
2
2
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 23 of 24
Computing with Fractions and Signed Numbers
Syllabus Objective: (2.1) The student will solve problems using operations on positive and
negative numbers, including rationals.
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
21
5
 2
8
8
7
2
5
3
9  25 

 
10  12 
2
4
9  25 
Example: 
 
10  12 

Multiply numerators and
denominators, determine
sign and simplify.
3  5
15
7
    1
2  4
8
8
Solving Equations Containing Fractions
Syllabus Objective: (3.5) The student will solve equations and inequalities in one variable
with integer solutions.
Strategy for Solving Equations: You solve equations containing fractions and signed numbers
the same as you do with whole numbers; the strategy does not change. To solve linear equations,
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.
1
2
Example: Solve: x    .
3
5
Undo adding one-third by subtracting onethird from both sides of the equation; make
1
2
6
x  
equivalent fractions with a common
3
5
15
denominator of 15.
1
1
5

 
3
3
15
11
x
15
Example: Solve:
x 2

5 3
Undo dividing by –5 by multiplying both
sides by –5. Cancel.
Multiply numerators and denominators,
determine sign and simplify.
x

5
 5   x 
 1   5  



2
3
2  5 
 
3 1 
10
1
x
or  3
3
3
Math 7, Unit 03: Applying Rational Numbers
Holt: Chapter 3
Page 24 of 24