Unformatted text preview: Fifth Grade
Fifth
Math Course I
Math
Fractions and Decimals 1 Fractions What are fractions?
They are not whole numbers
They
since they represent a part or a
portion of a whole.
portion
What part of the figure below is
What
shaded?
shaded?
One out of the three equal
One
sections is shaded. So we write
1/3 (onethird) of the whole
figure is shaded
figure
2 Fractions Every fraction has three parts.
Every
The top number is the
numerator, the bottom number is
numerator the
the denominator, and the middle
denominator and
line is the fraction line.
fraction
5 NUMERATOR (tells how many parts you have)
NUMERATOR 7 DENOMINATOR (tells the total number of parts)
DENOMINATOR Fraction line 3 Fractions What part of the
What
figure is
shaded?
shaded? 4 Proper Fractions Proper fractions have a value
Proper
that is less than 1, the
numerator is less than the
denominator
denominator
3
5 7
11 101
301 9
19
5 Improper Fractions Fractions that have a
Fractions
value that is equal to
1 or great than 1.
or
Four out of four
Four
sections are shaded.
4/4 (fourfourths) is
shaded.
shaded.
This represents the
This
whole object. So
4
=1
4
8
29
=1 6
=1
8
29 Improper Fractions
A 4
4 of figure A is
shaded and 3
4
of Figure B is
shaded. So a
total of 7 is
shaded. 4
7
So, = 1 3
4
4 We have one
We
whole object
and 3 of
4
another that is
shaded
shaded
7 B Fractions What part of the figure below is
What
shaded?
shaded?
What part of the figure is not
What
shaded?
shaded? 8 Fractions Which of the following fractions
Which
are proper fractions?
are Which of the following fractions
Which
are improper fractions?
are
12 , 7 ,
7
8 3,
4 9 , 15 , 10
6
16
3
9 Fractions If, on a true/false test, you get 17 correct
If,
answers out of 20 questions, a) what
fraction did you get correct? b) what
fraction did you get wrong?
fraction If, in a class of 30 students, 17 are
If,
females, a) what fraction are females? b)
what fraction are males?
what If 37 out of 100 golf balls found in a lake on
If
a golf course are orange, what fraction of
the golf balls are not orange?
the
10 Equivalent Fractions
3
Of the block is shaded
Of
4 6
Of the block is shaded
Of
8
The shaded sections represent the same amount of
The
3
6
each block. So those two fractions, 4 and 8 ,
represent the same amount of shaded area. They
have the same value.
have
6
3 and 6 are equivalent. 3 =
4
8
4
8
3 6 You can change 4 to 8 by multiplying its numerator
and denominator by 2. You can change 6 to 3 by
8
4
dividing its numerator and denominator by 1 Both
2.
1
procedures yield fractions that have the same value.
procedures Equivalent Fractions By multiplying the numerator and
By
denominator of a fraction by the
same number, you raise the fraction
to higher terms, getting an equivalent
fraction.
fraction.
x2 3
4 =
x2 x 10 x3
6
8 1
3 = 3
9 5
8 x3 =
x 10 12 50
80 Equivalent Fractions 3
8 Change
to a fraction with a
Change
numerator of 12.
numerator
3
Change 2 to a fraction with a
Change
denominator of 18.
denominator
5 = 35
11 = 33
6 ? 13 ? 4
7 = 16 1=
6 ?
54 2
9 =? 3=
8 24
13? ? 45 Reducing Fractions To reduce a fraction to lower
To
terms, you must find a number
terms you
other than 1 that divides evenly
into both the numerator and
denominator of the fraction. You
must continue to divide both the
numerator and denominator until
no whole number except 1
divides evenly into each.
divides
14 Reducing Fractions Reduce
÷2 30
=
42 30
42 15
21 ÷2
2 divides evenly into both
30 and 42 30
42 to lowest terms
÷3
15
=
21 5
7 ÷3
3 divides evenly into both
15 and 21 has been reduced to it’s
lowest terms 5 (prime numbers)
lowest
7
15 Reducing Fractions Reduce 12
24 ÷3
12
=
24
÷3 to lowest terms
÷2 4
8 =
÷2 ÷2
2
4 = 1
2 ÷2 16 Greatest Common Factor When reducing fractions, you
When
must find the greatest common
factor
factor Ex: GCF of 12 and 30 12 1 x 12
2 x 6 1, 2, 3, 4, 6 and 12 are factors of 12
3x4 30 1 x 30
2 x 15 1, 2, 3, 5, 6, 10, 15, and 30 are factors
1,
3 x 10
of 30
of
5x6
17 Greatest Common Factor Find the Greatest Common Factor
6 and 10 5 and 8 12 and 15 10 and 20 To reduce a fraction to its simplest
To
form, divide the numerator and
denominator by the Greatest
Common Factor
Common
12 1, 2, 3, 4, 6, 12 12 ÷ 3 = 4
15 1, 3, 5, 15
15 ÷ 3 18 5 Reducing Fractions Reduce to lowest terms
140 =
9=
910
12
100
150
36
90 = 18
30 = = 12
36 = 19 Mixed Numbers
Improper fractions have a value that is
greater than or equal to 1. Every improper
fraction can be expressed as either a
fraction
whole number or as the sum of a whole
number and a proper fraction (mixed
number
mixed
number)
number The fraction line not only separates the
The
numerator and denominator, it also implies
a division – the numerator divided by the
denominator
denominator Some improper fractions give whole
Some
number results
number
4 = 4 4 =1
15 = 3 15 = 5
4
3
20 Mixed Numbers 7
4 Some improper fractions give
Some
mixed number results
mixed
=4 7 19 = 5 19
5 3
= 1 R3 = 1 4 4
= 3 R4 = 3 5 Since we are dividing
Since
by 4, the remainder is
3
3 out of 4, or 4 . So
the mixed number is
3
1 and 4
Since we are dividing
Since
by 5, the remainder is
4 out of 5, or 4 . So
5
the mixed number is
4
3 and 5
21 Mixed Numbers Simplify
18
3 = 39
7=
29 =
7
22 Mixed Numbers & Improper
Mixed
Fractions
Fractions You can change a whole
You
number to an improper fraction
by putting the whole number
over a denominator of 1
15 = 15
1
100 = 100
1 7= 7
1
82
1 82 =
23 Converting mixed numbers
Converting
into improper fractions
into Consider how we changed
Consider
improper fractions into mixed
numbers
numbers
7
4 =4 7 3
= 1 R3 = 1 4 If you have a mixed number,
If
you can change it to an
improper fraction
improper
+
13
x4 = 4x1+3
4 = 7
4 24 Converting mixed numbers to
Converting
improper fractions
improper Change the following mixed numbers to
Change
improper fractions
improper
2
6
7 = 1
140 2 =
3=
54
1=
10
6
2=
122 3
25 Multiplying Fractions The easiest operation with
The
fractions is multiplication, since
to multiply fractions you just
have to multiply their
numerators and multiply their
denominators to get the
numerator and the denominator
of the answer.
of
1x3
2
5 = 1x3 = 3
2x5
10 26 Multiplying Fractions
6 x 42 = 6 x 42 = 252
15
63
15 x 63
945
Now we must reduce the answer
÷3
÷3
÷7
252
945 =
÷3 84
315 =
÷3 28
105 = 4
15 ÷7 The reducing process in that example was quite
The
involved since the numerator and denominator of
the fraction were large numbers. However, if we
reduced the fractions before we multiplied, the
solution would be a little easier.
solution
27 Multiplying Fractions
6 x 42
15
63
6
15 2
5 and 6 x 42
15
63 = Reduces to Therefore 42
63 reduces to 2x
5 2
3 = 3 x 14 = 3 x 14 =
42
70
25
70 x 25
1750
Reduce the
answer
answer ÷2 42
1750 =
÷2 ÷7
21
875 =
÷7 2
3 3
125
28 4
15 Cross Canceling in Fractions 3
70 14
25 The original fractions
and
are not
The
reducible. However, the product 42 can
750
be reduced to lower terms. What if1we
could reduce the fractions before we
before
multiplied? We can reduce the numerator
of the 2nd fraction with the denominator of
of
the 1st fraction.
the
1 3 x 14 = 3 x 1
70
25
5 x 25
5 = 3
125 This process is called cross canceling and
This
can ONLY be used when multiplying
fractions. You must divide the same
number evenly into any numerator and any
denominator.
denominator.
29 Cross Canceling
Ex: 2 x 18 x 15 = ?
10
26
2
18 9 15
x
x
1
10
26 Change 2 to the fraction 2 and reduce
1
18 to 9
10 5 1 2
x9
1
5
1 x 15
26 5 3
13 19
x
x 3 = 27
11
13
13
27 = 2 1
13
13 Cross cancel the 2 and the 26; cross
Cancel the 5 and the 15. Multiply the remaining fractions and
Change the improper fraction to a mixed
number 30 Multiplying Fractions
3
4 12
15 x x 20
7 = 8
25 x 75
100 = 31 Dividing Fractions To divide fractions, invert (find
find
the reciprocal of) the divisor (2nd
the
fraction) and change the division
into multiplication.
into
3÷2
5
1 = 3x
5 1
2 32 Dividing Fractions
2÷5
3
7 = 10 ÷ 7 =
15
27
35 ÷ 28 =
6
10 ÷ 45 =
3 33 Multiplying and Dividing Mixed
Multiplying
Numbers
Numbers To multiply and divide mixed
To
numbers, first change them into
improper fractions and then
multiply.
multiply.
Ex: 11 x 5 2 = ?
9
11 x 47 = 517
57 4
1
9=
9
9 33 Ex:
Ex: 4
5
15 x 16
4
3
1 4
1 x 51 =?
(Cross cancel)
3
5 x 4 = 20 = 20
=1
34
1
1 Multiplying and Dividing Mixed
Multiplying
Numbers
Numbers Solve the problems:
72
3 x 10 1 = ?
2 8 x 25 =?
16 10 2 ÷ 6 2
15
5 =? 95
6 =? ÷ 12 35 Fractional Parts of Numbers Fractions are often used to find a part of a number
Fractions
or quantity. You might encounter phrases such as
4
“ 5 of those surveyed”, “ 1 off the retail price”, or
3
“ 3 of the class.” In statements like these, you are
4
finding a part of a total amount. The word “of” in this
usage indicates multiplying the fraction and the
number.
number.
4
of 4725 means
of
5 4
5 1
of
3 of $630 means
3
of
4 of 36 means 1
3
3
4 x 4725
x 630
x 36
36 Fractional Parts of Numbers Solve these problems:
2
3
5
16 of 951 = ?
of 9 = ? Three fifths of those surveyed preferred “Colgate”
Three
toothpaste. If 100,000 people were surveyed, how
many preferred “Colgate” toothpaste?
many
1 A store is advertising a discount of 4 off all retail
store
prices. What would be the cost of a $304 item after
such a discount?
such
37 Adding and Subtracting Like
Adding
Fractions
Fractions “Like” fractions are fractions with
Like”
the same denominator
the
2
1
3
Ex: 5 + 5 = 5
Ex:
Adding and subtracting fractions
Adding
is straightforward. Leave the
denominator the same and add
or subtract the numerators.
or
38 Adding and Subtracting Like
Adding
Fractions
Fractions Solve these problems
2+3
7
7 +1
7 =? 52=?
6
6
5+2
8
8 +3
8 =? 63  11 + 8 = ?
25
25
25
39 Finding the Least Common
Finding
Denominator
Denominator Before you can add or subtract
Before
“unlike” fractions (different
denominators), you must first find the
Least Common Denominator (LCD)
Least The Least Common Denominator is
The Least
the smallest whole number that is
evenly divisible by the denominators
of other fractions
of
40 Finding the Least Common
Finding
Denominator
Denominator To find the LCD, find the
To
multiples of the numbers in the
denominator and choose the
lowest common number
lowest Ex:
Ex: 3+5
8
6 Multiples =? of: 6: 6, 12, 18, 24, 30, 36, etc
6:
24 8: 8, 16, 24, 32, 40, 48, etc
8:
24 41 Adding and Subtracting Unlike
Adding
Fractions
Fractions
3+1
4
5 4, 8, 12, 16, 20, 24, 28, … =? 5, 10, 15, 20, 25, 30, 35, … The LCD is 20, since 20 is the
The
smallest number that is divisible
by both 4 and 5. We have to
change 3 and 1 into 20ths
4
5
before we can add them.
before
3=?
4
20 x5 1= ?
5
20 x4 15
20
4
20 add 19
20
42 Adding and Subtracting Unlike
Adding
Fractions
Fractions
21=?
3
2
1+2
2
5 +9=?
10 3+5
4
6 +3
2  2
3 =? 43 Adding and Subtracting Unlike
Adding
Fractions
Fractions Find the Least Common
Find
Denominator.
Denominator.
Change each fraction to a
Change
fraction with the LCD as its
denominator.
denominator.
Add or subtract those like
Add
fractions.
fractions.
Reduce your answer and
Reduce
change any improper fraction
into a mixed number.
into
44 Adding and Subtracting Mixed
Adding
Numbers
Numbers 32
7 + 18 = ?
21 Change the mixed numbers to
Change
improper fractions
improper
The LCD is 21. Change each
The
fraction to a denominator of 32.
fraction
Add the like fractions, simplify
Add
and reduce.
and
3 2 = 23 =
7
7 69
21 + 1 8 = 29 =
21 21 29
21 add 98 = 4 14
42
21
21 =
3
45 Adding and Subtracting Mixed
Adding
Numbers
Numbers
21
12 +32
3  34
8
5 =? =? 46 ten
t hs
hun
tho dredth
usa
s
ndt
Ten
th hs
Hu ousa
ndr
n
ed dt hs
t
mil
lion housa
ths
ndt
hs Hu
n
Ten dred
Bill billio billion
ion ns
s
s
Hu
n
Ten dred
Mil milli millio
lion ons ns
Hu s
n
Ten dred
Th thou thous
ous sa an
and nds ds
s
Hu
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Ten dred
s
On s
es Reading and Writing Decimals 678, 310, 456, 127 . 3 4 1 0 1 9
(and) Any digit to the right of the ones place has
Any
a value that is less than one – a fraction
value
For example, in 0.576,
For
0.576
5
The 5 means 5 tenths 10
The 7 means 7 hundredths 7
100
The 6 means 6 thousandths 6 47
1000 Reading and Writing Decimals
Numbers to the right of the decimal point
Numbers
are read like whole numbers and are give a
value according to the position of its last
digit.
digit.
tenths
hundredths
thousandths 7
10 0.7 is read seven tenths 0.2 3 is read twentythree hundredths 23
100 0.1 7 7 is read one hundred seventyseven thousandths 1
48 77 1000 Reading and Writing Decimals If we have numbers to the right and left of
If
the decimal point, we really have a mixed
number.
number. Ex: 5.3 is read “five and three tenths” Ex: 16.08 is read sixteen and eight
Ex:
hundredths
hundredths 0.17 – write in words 426.9 – write in words Seven tenths = ? Fortyfive and six thousandths = ? Twelve hundredths = ?
49 Adding Decimals Place the numbers in a column so that their decimal
Place
points are line up.
points
Add the digits that have the same place value.
Place the decimal point in the answer below the
Place
other decimal points.
other Ex: To add 5.3 + .076 + 12.21
5.3
5.3
.076
.076
+12.21
17.586
17.586
If any of the numbers you are trying to add are
If
whole numbers without a decimal point, you can
simply place a decimal point at the end of the whole
number
number Ex: 26 = 26.
5 = 5.
5.
50 Adding Decimals 52 + 3.01 + 0.035 + 1.58 = ? 307.52 + 136 + .65 + 28 + 1.17 + 0.2 = ? 0.01 + 0.002 + 0.0003 + 0.00004 = ? 51 Subtracting Decimals Line up the decimal points before subtraction 27.973 – 2.241 = ?
27.973
 2.241
2.241
25.732
25.732 Subtraction is easier when both numbers have the
Subtraction
same number of decimal digits. We sometimes
have to give an alternate representation for a
decimal number.
decimal Ex: 0.7 can be written in different ways
0.70
0.70
0.700
0.700
0.7000
0.7000
Decimal points added after the decimal point do not
Decimal
change the value of the decimal number
change
52 Subtracting Decimals
.5762 – .34 = ?
.5762
.5762
 .3400
.3400
.2362
.2362
56.4 – 3.27 = ?
189 – 13.567 = ?
53 Multiplying Decimals To understand how to multiply
To
decimal numbers, we must look at
the fractions which the decimal
numbers represent.
numbers Ex: 0.3 x 0.71 = ?
3 and 0.71 = 71
0.3 =
100
10 3 So, 0.3 x 0.71 = 10 x
So, 71
100 213 =
= 0.213
0.213
1000 The number of decimal digits in the
The
product is the total of the number of
decimal digits in the numbers being
54
multiplied.
multiplied. Multiplying Decimals 19.86 x 0.089 = ?
1 9.86
9.86
x 0.089
17874
17874
15888
1.76754 has 2 decimal digits
has 3 decimal digits The answer has a total of
The
5 decimal digits
decimal
To place the decimal in the answer, start
To
from the right and move 5 places to the
left
left 43.6 x 2.7 = ?
6.075 x 3.14 = ? 55 Dividing Decimals If the divisor is a whole number, the decimal point is
If
placed in the answer above its position in the
dividend and the division is done ignoring the
decimal point
decimal 52.8 ÷ 4 = ?
52.8
13.2
13.2
4 52.8 Place the decimal point above the decimal
point in the dividend. If the divisor is not a whole number, you can make it
If
a whole number by moving its decimal point to the
right. You must also move the decimal point the
same number of places to the right in the dividend.
same 2.38 ÷ .7 = ?
2.38 .7
3.4
3.4
.7 2 . 3 8 1.
1.
2.
3.
4. Move the decimal point to change the
Move
divisor into a whole number
divisor
Move the point the same in the dividend
Place the decimal point in the answer
Place
above the moved decimal point
above
Do the division ignoring the decimal
Do
56
points
points Dividing Decimals 32.36 ÷ .04 = ?
32.36 .04 82.2 ÷ 6 = ?
82.2 57 Converting Fractions and
Converting
Decimals
Decimals To convert a fraction to a
To
decimal, remember that the
fraction line indicates division –
divide the numerator by the
denominator.
denominator. 3
=3÷4
4 .75
.75
4 3.00
3.00
58 Converting Decimals to
Converting
Fractions
Fractions To convert a decimal number to a
To
fraction, use the place value of the
digits to the right of the decimal point
– tenths, hundredths, thousandths,
tenthousandths, etc
tenthousandths,
9 Ex: 0.9 reads nine tenths
10 Ex: 0.35 reads thirtyfive
Ex:
hundredths 35
hundredths
100
2 Ex: 6.2 reads 6 and two tenths 6
Ex:
10
59 Converting Decimals and
Converting
Fractions
Fractions Write the fraction as a decimal
3
5 7
10 5
8 Write as reduced fractions or
Write
mixed numbers
mixed 0.32 5.375 76.8
60 ...
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 Winter '08
 Staff
 Decimals, Fractions, Decimal, Elementary arithmetic, Greatest common divisor

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