Download Method 3: Convert Fractions to Decimals

SS4.1 Decimal Notation, Order and Rounding
Place Values
Recall the place values that we discussed in Chapter 1.
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Traveling left from the one’s place, each place value increases by a factor of 10.
1 x 10 = 10
10 x 10 = 100
100 x 10 = 1,000
1000 x 10 = 10,000
10,000 x 10 = 100,000
100,000 x 10 = 1,000,000
Similarly, if we travel right form the millions place, each place value decreases by a
factor of 10.
1,000,000  10 = 100,000
100,000  10 = 10,000
10,000  10 = 1,000
1,000  10 = 100
100  10 = 10
10  10 = 1
Recall also that immediately to the right of the ones place is the decimal point. The
decimal separates whole numbers from fractional parts of numbers. If we continue with
our pattern of division from one, we will see the following fractions begin to form. The
way that we read the fraction is where the place values below the decimal get their
names.
75
1  10 = 1/10 (read one tenth) Thus
tenths place
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/10  10 = /100
(read one hundredth) Thus hundredths place
1
1
/100  10 = /1000 (read one thousandth) Thus thousandths place
This pattern continues and we see the pattern is similar to the pattern as we travel to the
right from the ones place.
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You read numbers with decimals the same way that you read a whole number except that
when you come to the decimal you include the word and and once you have read the
entire number say the name of the place value farthest to the right of the decimal.
Example:
Read & write the following
a) 107.2
b) 2,073.52
c) 2.7329
d) 3.338173
Just as with whole numbers, we can write decimals in expanded notation. Recall the
following is standard form to expanded form.
278 = 200 + 70 + 8
76
We are going to take writing a decimal number in expanded form in small pieces. First
we are going to learn to write a decimal number as a fraction and then how to write just a
decimal in expanded form and finally we will learn to write a number that contains both
whole numbers and decimals in expanded notation.
Method 1: Writing a Decimal as a Fraction
Step 1 Read the decimal as we learned by saying the number and then the place value
of the number farthest to the right
Step 2 Use the place value name as the denominator
Step 3 Write the number to the right of the decimal as the numerator
Example:
Write the following as fractions
a) 0.2
b) 0.24
c) 0.592
d) 0.6217
e) 0.0218
Another way of looking at converting a decimal to a fraction is to consider the number of
places to the right of the decimal. The number of places will indicate the number of zeros
the factor of 10 in the denominator will need. (This is simply the concept that when dividing by
factors of 10 we move the decimal to the left the same number of times as the number of zeros in the factor
of ten!)
Method 2: Writing a Decimal as a Fraction
Step 1 Count the number of places to the right of the decimal
Step 2 Use decimal number without regards to the decimal as the numerator
Step 3 Make the denominator the factor of 10 that has the same number of zeros
counted in step 1.
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Example:
Convert the decimals to a fraction by using method 2
a) 0.39
b) 0.00358
c) 1.085002
Writing a Decimal in Expanded Form
Step 1 Write the number just to the right of the decimal a fraction as described above,
ignoring all numbers to the right of it.
Step 2 Add to that the next number written as a fraction while ignoring all other
numbers
Step 3 Continue step 2 until the last number to the right is added
Example:
Write the following in expanded form
a) 0.24
b) 0.592
c) 0.6217
d) 0.048
Remember that we could expand 1 ½ to 1 + ½. We have just seen that decimals can be
written as fractions so a whole number is added to the decimal parts in expanded form.
Example:
Write the following in expanded form
a) 289
b) 12.89
c) 102.039
78
The concept of writing a decimal in expanded form can be used in reverse to write a
mixed number or a fraction as a decimal. (At this time only numbers that have a fractional portion
with a denominator that is a factor of 10 will be considered.)
Converting Mixed Numbers/Fractions to Decimals (denom. factor of 10)
Step 1 Count the number of zeros in the factor of 10 in the denominator
Step 2 Write down the numerator number
Step 3 Place the decimal by counting the number of spaces, from the furthest right
number, as indicated by step 1. If the number is a mixed number, write the
whole portion to the left of the decimal place.
Example:
Write the following as a decimal
a) 15 3/10
b)
275
/10
c) 152 17/1000
d) 15,303/10,000
When comparing the value of two decimals there are two ways of making a comparison.
The first is to change the numbers to fractions and compare the fractions as we did in the
last chapter. (Recall that this was finding the cross products and the larger is the larger number.) The
second method is to compare each digit, from left to right, until you find the larger digit,
in which case you have found the larger number.
Method 1: Comparing Decimals using Fractions
Step 1 Change decimals to fractions
Step 2 Find the cross products
Step 3 Compare the cross products (the larger is the larger number)
(Of course, I believe that it goes without saying that the larger whole number is the
larger number, with no need for comparison of decimal portions)
Example:
Use < or > to compare the following decimals using method 1
a) 0.389  0.385
b) 2.0719
c) 5.72


2.0772
2.1
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Method 2: Comparing Decimals using Number by Number Comparison
Step 1 Compare the first numbers of each decimal (furthest left)
Step 2 Continue the comparison working right, number by number
Step 3 When a larger number is encounter, the larger decimal number has been found
Example:
Using < or > compare the decimal numbers
a) 0.253  0.25301
b) 2.535

c) 0.985 
2.513728
0.9850001
Rounding Decimals
When rounding a decimal it is done in the same exact way that it is done with whole
numbers, except that we now have decimals.
Step 1 Locate the place value that you are rounding to
Step 2 Look at the number to the right of this place
If the number is greater than or equal to 5 then we make the number in our
place value one larger
If the number is less than 5 then we have no change in the number in our
place value
Step 3 Truncate the new number (make everything to the right of the place value zero)
Example:
Round to the nearest one
a) 754.1
b) 635,806.61
Example:
Round to the nearest tenth
a) 6,841.15
b) 2.013
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Example:
Round to the nearest thousandth
a) 1.90358
b) 0.738599233809357
HW
p. 201-202 #2-96 evens
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SS 4.2 Addition & Subtraction with Decimal Notation
There is not much to say about adding and subtracting decimals. The most important
thing to remember is that the place values must be lined up in order to add and subtract,
just as with whole numbers. The easiest way to do this with decimals is to line up the
decimal points.
Adding / Subtracting Decimals
Step 1 Line up the decimals
Step 2 Fill in with zeros to the right where needed
Step 3 Add or Subtract as normal
Example:
Add or Subtract as indicated
a) 1.902 + 37.2
b) 293.001 + 26.12
c) 2.0201 + 35.12
d) 13.7  2.079
82
e) 26.85  5.9
f) 109.001  2.79892
g) 5.82 + (2.7  1.09)
h) 17.01  (2.3 + 1.735 + 0.00352)
Example:
Previously I made $7.34 per hour. I now make $10.61 per hour.
How much more do I make now?
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Example:
Marge goes to the store and buys a ham for $12.19, a can of
pineapple for $1.84, and a bottle of whole cloves for $2.39. How
much does she spend?
Example:
If Marge gives the clerk a $50 bill how much change will she get?
Example:
If I write the following checks and I have $259.73 in my checking
account, how much will be left?
Gas -- $72.81
PG&E -- $54.17
Phone -- $48.46
HW
p.207-210 #2-60 even & #72, 74, 76, 78
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SS 4.3 Multiplication with Decimal Notation
Multiplying Decimals
Step 1 Multiply as usual ignoring the decimals
Step 2 Count up the total number of places to the right of the decimals
Step 3 Put the same number of decimals from step2 into the answer by counting from
the right
Example:
Multiply
a) 0.2 x 0.3
b) 1.1 x 2.1
c) 3.21 x 3
d) 0.001 x 0.2
e) 0.05 x 0.05
f) 15.1 x 0.002
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Multiplication by Factors of 10
Recall that factors of 10 add zeros to a number. We can also relate this as moving the
decimal place. If we multiply by a factor of 10 that is 10 or larger then the decimal is
moved to the right the number of times indicated by the number of zeros. If multiplying
by a fractional factor of 10 then the decimal moves to the left the number of times
indicated by the number of zeros in the denominator factor of ten, since this is division.
Example:
0.02532 x 100
Example:
17.3013 x 1,000
Example:
0.1932 x 1/10
Example:
152.0198 x 1/1000
(i.e. Division by 10 – recall that division is
multiplication by the reciprocal)
When we see larger numbers mentioned in newspaper articles and magazines, they are
often discussed using decimals. For instance, an article may say that there are 32.6
million people living in the Americas. This notation is the same as saying 32.6 x
1,000,000. We can relate multiplication by such larger numbers in a simpler manner by
using exponents to indicate the number of zeros that the factor of ten has. To accomplish
this short hand, we write the number 10 and then use an exponent to tell us how many
zeros the factor of ten has. Here is the same example as above written in this manner.
32.6 x 106 people
86
Money Conversion
The relationship between dollars and cents in the US is a factor of 102. To make the
conversions we need only divide or multiply by 100.
Converting Dollars to Cents
Step 1 Multiply by 100 (Move the decimal 2 places to the right)
Step 2 Attach a cent sign
Example:
Write the following as cents
a) $52.12
b) $0.65
Converting Cents to Dollars
Step 1 Divide by 100 (Move the decimal 2 places to the left)
Step 2 Attach a dollar sign
Example:
Write the following as dollars
a) 289
b) 32
Here are some word problem examples to keep you entertained!
Example:
If Gas Station #1 charges $1.71 per gallon and Gas Station #2
charges $1.97 per gallon and you fill up your 10 gallon tank, how
much would you save a Gas Station #1?
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Example:
If it costs $0.35 to place a call and $0.05 per minute after the first
minute, how much would it cost to make a 6 minute call?
HW
p. 217-218 #2-66 even
88
SS 4.4 Division with Decimal Notation
Division with decimals is a little different than division with whole numbers because we
must deal with the decimals!
Division of Decimals
Step 1 Move the decimal out of the divisor by moving it the same number of places in
the dividend
Step 2 Bring the decimal up into the proper place in the quotient (Can be done last too)
Step 3 Divide as usual adding zeros until there is no remainder or until you can round to
an appropriate place value
Step 4 Check
Example:
Divide these decimals by whole numbers
a) 18.21  3
b) 18.75  5
Example:
Divide these whole numbers by whole numbers and write the
answers using decimals
a) 17  5
b) 287  25
89
Example:
Divide the decimals or whole numbers by decimals
a) 18  0.02
b) 1.29  0.5
Let’s also try some word problems. When you get an answer for a word problem,
sometimes rounding the answer is necessary. When talking about money, for instance, it
is not appropriate to get an answer such as $25.0001, since money is only written to the
nearest hundredth. It is usually not considered good practice to get an answer with more
place values to the right of the decimal than the smallest number of decimal places.
Example:
It costs farmer Bill $278.15 to fee his 7 pigs for a month. How
much does it cost to feed 1 pig for a month?
Example:
On my last tank of gas I traveled 489 miles. If my tank holds 11
gallons, how far can I drive on one gallon of gas?
90
Order of Operations with Decimals
Order of Operations with decimals is no different than when dealing with whole numbers.
You need only remember the order of operations (PEMDAS) and how to work with
decimals.
Example:
4  0.4 + 0.1  5  0.12
Example:
4.2  5.7 + 0.7  3.5
Example:
The average global temperatures over four years in degrees
Fahrenheit were as follows: 59.85, 59.74, 59.23, and 59.36. What
was the average of the average global temperatures over these four
years?
HW
p. 227-228 # 2 – 42 even, # 50 – 78 even
91
SS 4.5 Converting from Fractional Notation to Decimal Notation
Your author suggests that one method of converting fractions to decimals is
building a higher term, such that the denominator is a multiple of 10. This method is a
poor one as it is time consuming and it only works for current denominators that are
factors of multiples of 10. In short, this method is not one that I recommend, and I will
not discuss it further.
Fractions to Decimals
Step 1 Divide numerator by denominator, placing a decimal after the whole number and
adding zeros
Step 2 Divide until one of the following happens
a) an answer is found (there is no remainder) – called a terminating decimal
b) until a pattern becomes apparent in the quotient – called a repeating decimal
c) until you reach an appropriate place to round without finding any sign of a
pattern appearing – usually appropriate is the same number of decimals as in
other decimals involved in a problem.
Example:
a)
Change the following to decimals
¼
b)
2
/5
c)
1
/3
d)
1
/9
e)
2
/3
92
2
f)
g)
7
/11
/97
Repeating decimals are easy to spot and easy to deal with by simply putting a line over
the repeating portion of the decimal when we are merely giving a decimal representation
of the fraction. However, when we need to work with a decimal in an applied problem
we need to know what to do with the decimal. In applied problems we need to round a
repeating decimal to an appropriate number of decimal places and use it to do the
appropriate calculations. The appropriate place to round for applied problems is as
follows:
How to Round Repeating or Non-Terminating Decimals for Applied Problems
Round to the same place value as the largest place value in the problem.
Solving applied problems is not difficult, but there are multiple methods for doing
calculations. I will indicate the 3 methods, and list them in order of preference, according
to the most accurate answer achieved. The biggest problem with problems of mixed form
is accuracy. Because rounding causes some error it can prevent us from achieving the
best answer to a problem.
Method 1: As Is
Step 1:Put all decimals over one & change all mixed numbers to improper fractions
Step 2:Carry out the calculations as you normally would using order of operations
Step 3:If the final answer has a decimal in either the numerator or denominator, use
division to change the answer to a decimal and if it is a proper fraction, simply
reduce when necessary and leave it alone.
Example:
a)
Calculate
6.84  2 ½
93
b)
3.375  5 1/3
At this time it is useful to memorize some decimal to fraction conversions. The
following table is a list of very helpful conversions to memorize.
Fraction
Decimal
½
0.5
1
/3
0.333
2
/3
0.666
¼
0.25
¾
0.75
1
/5
0.2
2
/5
0.4
3
/5
0.6
4
/5
0.8
1
/6
0.1666
5
/6
0.8333
1
/8
0.125
3
/8
0.375
5
/8
0.625
7
/8
0.875
1
/9
0.111
2
/9
0.222
4
/9
0.444
5
/9
0.555
7
/9
0.777
8
/9
0.888
Method 2: Treat Decimal as a Fraction
Step 1:Convert decimals to fractions by methods discussed in section 1 of this chapter or
by using its memorized conversion.
Step 2:Do the appropriate operations using order of operations
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Example:
a)
b)
Calculate
384.8  4/5
1/3  2.5
Method 3: Convert Fractions to Decimals
Step 1:Convert fractions to decimals using this sections’ methods or using memorized
conversions.
Step 2:Use order of operations to complete the problem
Example:
a)
b)
Calculate
6.84  2 ½
1
/9  0.875
95
*Note: A problem occurs in using this method when fractions are represented as
repeating decimals. When a fraction is a repeating decimal it is better to use method 2.
Let’s use method 2 on this problem and compare our answers. Change 0.875 to a
fractions and then do the calculation. The decimal representation is then needed to make
a comparison.
1
/9  0.875
Now, we get to the more difficult problems, where we may need to combine methods, or
give the problem thought before we jump in using a particular method.
Example:
a)
b)
Calculate using order of operations
7
/8  0.86  0.76  ¾
5
/6  0.0765 + 5/4  0.1124
96
c)
7
/8  0.86  0.76  ¾
d)
5
/6  0.864 + 14.3  8/5
HW
p. 233-234 #2-36 even, #50-78 even
97
SS 4.6 Estimating
Estimating can be used in the following situations
a) checking an answer
b) finding an approximate price
Loose rules for estimating
a) keep 1 or 2 non-zero digits rather they are in the wholes or decimals
Example:
A color TV -- $299
CD System -- $249.95
Entertainment Center -- $109.95
a) What is the approximate cost of a CD System & an Entertainment
center?
b) What is the approximate cost of 9 TV’s?
c) About how much more does a TV cost than a CD System?
98
d) About how many TV’s can you buy for $1700?
Example:
Round to the nearest tenth & estimate
0.02 + 1.31 + 0.34
Example:
Round to the nearest one & estimate
12.9882  1.0115
Example:
Round to so that there are 1 or 2 nonzero rounding digits
a) 3.6  4
99
b) 49  7.89
*Remember that the idea is to make the problem easier to calculate without totally
destroying the accuracy!
HW
p. 239-240 #2-36 even
100
SS 4.7 Applications and Problem Solving
I do not have any additional comments to add to this section. We have been over word
problem strategies since the first days of class, and I don’t think that I have anything
more that I need to say. Instead, let’s just work some problems of varying types that
come directly from the book.
Example:
What is the cost, in dollars, of 17.7 gallons of gasoline at 119.9
cents per gallon? (The book indicates to round to the nearest cent,
but it should not be necessary to remind you that a problem that
involves money as the answer needs to be rounded to the nearest
cent. You should already be aware of that fact!!)
Example:
In Texas, one of the state lotteries is called “Cash 5.” In a recent
weekly game, 6 winners shared the lottery prize of $127,315
equally. How much was each winner’s share?
101
Example:
A family checked the odometer before starting a trip. It read
22,456.8 and they know that they will be driving 234.7 miles.
What will the odometer read at the end of the trip?
Example:
Peggy filled her van’s gas tank and noted that the odometer read
26,342.8. After the next filling, the odometer read 26,736.7. It
took 19.5 gallons of gas to fill the tank. How many miles per
gallon did Peggy’s van get?
102
Example:
The average video game costs 25 cent and runs for 1.5 minutes.
Assuming a player does not win any free games and plays
continuously for an hour, how much money, in dollars does it cost?
Example:
0.8 cm
d
3.91
cm
Find the length of d in the above figure. Assume that both edges are 0.8 cm wide.
HW
p. 249-254 #6-54 mult. of 6 & #62-72 even
103