1
3.1
Place Value and Word Names of Decimals
The word decimal is derived from the Latin word decem, which means “ten.” Our real
number system is based on powers of 10. Thus, we refer to it as the decimal system of
numeration. Decimals are numbers which may be expressed as a fraction with a denominator
that is some power of 10. We commonly refer to the decimal part of a number as the part
of the number to the right of the decimal point. An example is given below.
45.489
.489 is the decimal part.
Decimal parts are always equal to some proper fraction that has a denominator that is some
power of ten. Examples are given below.
0.4 is equal to 4 .
10
0.45 is equal to 45 .
100
0.489 is equal to 489 .
1000
0.4556 is equal to 4556 .
10,000
In general, any real number containing a decimal point is referred to as a decimal. Each
decimal may be thought of as a mixed number. The part of the number to the left of the
decimal point is the whole number part, and the part to the right of the decimal point is equal
to some fraction with a denominator that is a power of ten.
MATH FACTS ABOUT DECIMALS
Every decimal may be written as a fraction or mixed number.
The decimal point acts as the + sign between the whole number part
and the decimal part.
Each decimal part may be written as a fraction with a denominator
equal to a power of ten.
Example:
3.45 = 3 + 0.45 = 3 +
45
45
= 3
.
100
100
2
Writing Decimals as Fractions or Mixed Numbers
Any decimal may be written as a fraction or a mixed number where the fraction has a
denominator that is a power of ten. The number of places to the right of the decimal point
indicates the power of ten in the denominator of the fraction. A technique for writing a
decimal as a mixed number is summarized in the following procedure.
PROCEDURE TO CONVERT A DECIMAL INTO A FRACTION OR MIXED NUMBER
1.
If the decimal contains a nonzero part to the left of the decimal point,
rewrite the decimal as a whole number added to the decimal part.
The decimal point acts as the + sign.
2.
Write the decimal part over a power of ten as a fraction. The number
of places to the right of the decimal point will be equal to the power
of ten in the denominator of the fraction.
3.
Remove the + sign to write the sum as a mixed number.
Example 1
Write 12.099 as a mixed number.
12.099 = 12 + 0.099
3 places
Since there are 3 places to the right of the decimal point, the fraction will
have a denominator of 103 or 1000.
12 + 0.099 = 12 + 0993 = 12 + 99
10
1000
This is equal to the mixed number 12 99 .
1000
Note that the left zero in the number 099 was dropped.
3
Example 2
Write 1.00870005 as a mixed number.
1.00870005 = 1 + 0.00870005
8 places
Since there are 8 places to the right of the decimal point, the fraction will
have a denominator of 108 or 100,000,000 .
1 + 0.00870005 = 1 +
00870005
100,000,000
=1+
870,005
100,000,000
This is written as the mixed number, 1 870,005
100,000,000
Example 3
.
Write the decimal 0.04507 as a fraction.
In this example, there is no whole number part. Thus, this decimal will
consist solely of a proper fraction. Since there are 5 places to the right of the
decimal point, the fraction will have a denominator of 105 = 100,000.
0.04507 = 04507 =
100,000
4507
100,000
Place Value
The place values of decimal parts are tenths, hundredths, thousandths, and other fractional
parts with denominators that are powers of ten. For example, the place value of 7 in 0.78 is
tenths and the place value of 8 is hundredths. If 0.78 is written as a fraction, we obtain
0.78 = 78 .
100
78
however, is equal to 70 + 8 = 7 + 8 .
100
100 100
10
100
In summary, a simple technique to identify the place value of a decimal digit consists of
counting the number of places to the right of the decimal point that the digit is located. The
number of places is equal to the power of 10 in the denominator of the fraction that
represents this digit. This procedure is summarized on the following page.
4
PROCEDURE TO IDENTIFY THE PLACE VALUE OF A DECIMAL DIGIT
1.
Count the number of places to the right of the decimal point that the digit is
located.
2.
The number of places is equal to the power of 10 in the denominator of the
fraction that represents this digit.
Example 4
Find the place values of the digits 9 and 2 in 0.93712 .
The digit 9 is one place to the right of the decimal point. Thus, the fraction
that represents this digit is 1 1 = 1 .
10
10
The digit 9 is in the tenths place and the value of 9 is nine tenths.
The digit 2 is five places to the right of the decimal point. Thus, the fraction
1
that represents this digit is 1 5 =
.
10
100,000
The digit 2 is in the hundred-thousandths place and the value of 2 is two
hundred-thousandths.
Example 5
Find the place values of each of the digits in 0.539 and write this decimal as
a sum of three fractions.
The digit 5 is one place to the right of the decimal point. Thus, 5 is in the
tenths place and the value of 5 is five tenths or 5 .
10
The digit 3 is two places to the right of the decimal point. Thus, 3 is in the
hundredths place and the value of 3 is three hundredths or 3 .
100
The digit 9 is three places to the right of the decimal point. Thus, the 9 is in
the thousandths place and the value of 9 is nine thousandths or 9 .
1000
The decimal 0.539 may be written as 5 +
10
3
100
+
9
.
1000
5
Word Names of Decimals
The word name of a decimal may be obtained by writing the mixed number form of the
decimal and then writing the word name of the mixed number. This procedure is given here.
PROCEDURE TO WRITE THE WORD NAME OF A DECIMAL
1.
Write the decimal as a mixed number.
2.
Write the word name of the mixed number. Be careful to only use
the word “and” between the names of the whole number and the
fraction.
Example 6
Write the word name of 45.067 .
Since there are 3 places to the right of the decimal point, the mixed number
is equal to
45 + 673 = 45 + 67
= 45 67 .
1000
10
1000
The word name of this mixed number is forty-five and sixty-seven
thousandths.
Example 7
Write the write word name of 0.7000 .
There are 4 places to the right of the decimal point. Thus, this decimal is
equal to 7000
= 7000 .
4
10
10,000
The word name is seven thousand ten-thousandths.
MATH FACT
When converting a decimal into a mixed number or a fraction, the
number of zeroes in the denominator of the fraction will always be
equal to the number of places to the right of the decimal point.
Example:
3.405 = 3 405
1000
3 places = 3 zeroes
6
3.2
Listing Decimals in Order and Rounding
Adding or Deleting Zeros From the Far Right Side of a Decimal
If you write the decimals 0.8, 0.80, and 0.8000 as fractions you obtain
0.8 =
8
10
0.80 =
80
100
0.8000 =
8000
.
10,000
All of these fractions reduce to 8 and thus 0.8 = 0.80 = 0.8000 . Because the addition of
10
the extra zeroes at the far right side results in an equivalent fraction, the value of the
decimal is not changed. Likewise, deleting zeroes from the far right side of a decimal
does not change the value of the decimal.
MATH FACT
Adding or deleting zeroes from the far right side of a decimal will not
change the value of the decimal.
Example:
0.09000 = 0.09 = 0.09000000
Note: Only zeroes on the far right can be added or deleted. In the
example given here, the zero to the left of 9 can not be deleted,
nor can a zero be added to the left of 9.
Example 1
Write 0.07 as an equivalent decimal with three, four, five, and six places to
the right of the decimal point.
0.07 = 0.070 = 0.0700 = 0.07000 = 0.070000
Listing Decimals in Order of Value
When comparing fractions, it is necessary to write all of the fractions as equivalent fractions
with a common denominator. If we wish to list the decimals 0.42, 0.4102, 0.411, and 0.401
in order from least in value to greatest in value, we also compare the fraction equivalents of
each of the decimals. To make sure that the decimals have fraction equivalents with the
same denominators we first add zeroes to the far right of the decimals to make them the same
length and then compare the fraction equivalents of the decimals. This procedure is shown
on the following page.
7
We add two zeroes to 0.42, one zero to 0.411, and one zero to 0.401 . Then we compare the
fraction equivalents.
0.42 = 0.4200 =
0.4102 =
4200
10,000
4102
10,000
0.411 = 0.4110 =
4110
10,000
0.401 = 0.4010 = 4010
10,000
The list of fractions, in order from smallest in value to largest in value is
4010
, 4102 , 4110 , 4200 .
10,000
10,000
10,000 10,000
Thus, the list of decimals, ordered from smallest to largest in value is
0.401, 0.4102, 0.411, 0.42 .
The procedure for comparing values of decimals is summarized here.
PROCEDURE FOR COMPARING DECIMALS
1.
Add zeroes to the far right of the decimals so that they all have the
same number of places to the right of the decimal point.
2.
Compare the fraction equivalents of the decimals.
Example 2
List the decimals 1.14, 1.1088, 1.149, and 1.1503 in order from smallest to
largest in value.
Rewrite all of the decimals so that they have 4 decimal places. Add zeroes
to the decimals and then write each as its equivalent mixed number. This is
done on the following page.
8
1.14 = 1.1400 = 1 1400
(second smallest)
1.1088 = 1 1088
(smallest)
10,000
10,000
1.149 = 1.1490 = 1 1490
(third smallest)
1.1503 = 1 1503
(largest)
10,000
10,000
Comparing these mixed numbers, we see that the smallest decimal is 1.1088,
the second smallest is 1.14, the third smallest is 1.149, and the largest is
1.1503 . The list from smallest to largest is 1.088, 1.14, 1.149, 1.1503 .
Rounding Decimals
To round a decimal, use nearly the same procedure that was used to round whole numbers
in Chapter One. This procedure is given here.
PROCEDURE TO ROUND A NUMBER
1.
Locate the digit to be rounded.
2.
Look at the digit to the right. If the digit to the right is greater than or
equal to 5, then add 1 to the digit that is rounded. If the digit to the
right is less than or equal to 4, then leave the first digit as it is.
3.
When rounding decimal places to the right of the decimal point, do
not include any digits to the right of the rounded digit.
Example 3
Round 3.40522 to the nearest hundredth.
Add 1 to 0
3.40522 rounds to 3.41
Look at 5
The digit in the hundredths place is 0. Look at the digit to the right which is
5. Since 5 is greater than or equal to 5, we add 1 to 0 and round up to 3.41,
and we do not include any digits to the right of the rounded digit.
9
Example 4
Round 2.999804 to the nearest thousandth.
Add 1 to 9
2.999804 rounds to 3.000
Look at 8
The digit in the thousandths place is the third 9 to the right of the decimal
point. Since the next digit 8 is greater than or equal to 5, we add 1 to 9.
Adding 1 to 9 results in 2.999 rounding up to 3.000 . In effect, the following
addition is performed.
2.999
+.001
Note that the digits beyond the rounded digit are dropped.
3.000
Example 5
Round 104.032 to the nearest tenth and also round to the nearest hundred.
To round to the nearest tenth, we look at the digit to the right of 0 in the
tenths place. Since 3 is less than 5, nothing is added to 0 and the digits to the
right of zero are dropped.
Leave 0 as it is
104.032 rounded to the nearest tenth is 104.0
Look at 3
To round to the nearest hundred, we look at the digit to the right of 1 in the
hundreds place. Since 0 is less than 5, nothing is added to 1 and zero
placeholders are inserted for the ones and the tens places. No decimal part
is included.
Leave 1 as it is
104.032 rounded to the nearest hundred is 100.
Look at 0
MATH FACT
When rounding to a whole number place, never include a decimal point or
any decimal places.
10
Rounding and Measurement
When the decimal 12.3004 is rounded to the nearest thousandth, the result is 12.300 . One
might say that we could write this result as 12.3, but this would be incorrect when 12.3004
represents a measurement. The rounded result of 12.300 implies that the measurement is
accurate to a thousandth, however 12.3 implies that the measurement is accurate only to the
nearest tenth. For example, if a chemistry student obtained the mass of a compound on an
electronic balance as 12.3004 grams and rounded this result to a thousandth of a gram, they
would report this mass as 12.300 grams. On the other hand, if a contractor used 12.25 cubic
yards of cement on a job and rounded this result to the nearest tenth of a cubic yard, they
would report this amount as 12.3 cubic yards. These two amounts (12.300 and 12.3) are
equivalent, yet the extra zeroes in 12.300 imply a greater degree of precision.
Example 6
A fuel oil truck delivers fuel oil to you through a hose running from the back
of the truck. In a heating season your fuel oil tank is filled three times: once
with 180.8 gallons of oil, once with 121.32 gallons of oil, and once with
about 85 gallons of oil. If you were calculating the total amount of heating oil
used, would you report 387.12 gallons, 387.120 gallons, 387.1 gallons, or 387
gallons?
Since these measurements, especially the measurement of 85 gallons, are not
very precise, it would be appropriate to report this sum as 387 gallons.
Example 7
In a chemistry experiment, you weigh out 3.7000 grams of a substance on a
state-of-the-art balance and then divide this amount into two equal amounts
of 1.8500 grams. Should you report this as 1.85 grams, 1.9 grams, or 1.8500
grams?
Since the measurement device in this example is very precise, you would
want to report this mass as 1.8500 grams.
11
3.3
Operations with Decimals
Addition and Subtraction of Decimals
The process of adding and subtracting decimals consists of placing the decimals over each
other so that the decimal points line up, adding zero placeholders at the far right so that all
of the decimals have the same number of decimal places, and then adding or subtracting from
right to left as if the decimals were whole numbers. This procedure is summarized here.
PROCEDURE FOR ADDING OR SUBTRACTING DECIMALS
1.
Place the decimals in column form so that the decimal points line up.
2.
Add zeroes to the far right so that all of the decimals have the same
number of places to the right of the decimal point.
3.
Add or subtract from right to left as if the decimals were whole
numbers. Insert the decimal point in the answer in the same location
as in the decimals added or subtracted.
Example 1
Add 2.3 + 4.909 + 1.01
Place these decimals in column form so that the decimal points line up.
Then, add zeroes to the far right of the decimal points to make all of the
decimal parts the same length.
2.3
4.909
+ 1.01
2.300
= 4.909
+ 1.010
Now, add these numbers in the same way you would add 2300+4009+1010.
1
2.300
4.909
+ 1.010
8.219
Note that when the placeholders 3 and 9 are added, the
result is 12 tenths. 2 is written in the tenths placeholder
location and 1 (ten tenths) is carried to the ones column.
12
Example 2
Subtract 3.6
2.556 .
Place these decimals in column form so that the decimal points line up.
Then, add zeroes to the far right of 3.6 so that both decimals have the same
number of places.
3.6
2.556
3.600
2.556
Now subtract these numbers in the same way that you would subtract the
whole numbers 3600 2556 .
9
5 1010
3.600
2.556
Note that the same borrowing procedure is used as in the
subtraction of 3600 2556.
1.044
Example 3
Add 3 + 2.08 + 6.1 .
In this example, 3 is not a decimal but a whole number. 3 can be written as
the decimal 3.00 . Also, 6.1 can be written as 6.10 .
3.00
2.08
+ 6.10
The addition is performed in the same way as the
addition 300 + 208 + 610.
11.18
MATH FACT
A whole number may always be written as an equivalent decimal by
adding a decimal point to the right of the ones place and adding as
many zeroes as desired to the right of the decimal point.
Example:
25 = 25.0 = 25.0000
13
Multiplying Decimals
When we multiply 1 2 × 3 5 we convert the mixed numbers into improper fractions and
100
multiply to obtain the result
10
102
35
3570
×
=
.
100
10
1000
Since 1 2 = 1.02 and 3 5 = 3.5, we could say that 1.02 × 3.5 = 3570 . But 3570 is
100
10
1000
1000
570
equal to the mixed number 3
which in decimal form is 3.570 . Thus, 1.02 × 3.5 = 3.570.
1000
In this example, the decimals multiplied contained 2 decimal places and 1 decimal place.
The answer contained 2 + 1 or 3 decimal places. Likewise, in the fraction multiplication, the
fractions contained a denominator that was ten to a power of 2 and ten to a power of 1. The
fraction answer contained a denominator of ten to a power of (2 + 1) = 3. The final answer
of 3.570 could have been obtained by multiplying 1.02 × 3.5 as if they were whole numbers
and placing the decimal point so that the answer contained 3 places. A general procedure for
decimal multiplication is given here.
PROCEDURE FOR DECIMAL MULTIPLICATION
1.
Multiply the decimal numbers as if they were whole numbers.
2.
Count the number of decimal places in each factor, and add these
numbers to find the sum of the decimal places. The answer will have
a number of decimal places equal to the sum.
Example 4
Multiply 3.05 × 2.012
Multiply 3.05 × 2.012 as you would multiply 305 × 2012. This result is
613660. We must insert the decimal point in this answer so that there are 5
decimal places.
3.05
× 2.012 = 6.13660
2 places + 3 places = 5 places
2.012
× 3.05
10060
603600
6.13660
14
Example 5
Multiply 5 × 4.2 × 3 .
This multiplication is performed in the same manner as the multiplication 5
× 42 × 3. The total number of decimal places in the answer is only 1 since
the whole numbers 5 and 3 have no decimal places.
5
×
4.2
× 3
63.0
0 places + 1 place + 0 places = 1 place in answer
MATH HINT
When multiplying decimals, first estimate the answer if possible.
Estimate by rounding the decimals to whole numbers and then
multiplying. Compare your estimate to the actual answer.
Example 6
Estimate 2.02 × 5.01. Then multiply the numbers out to find the actual
answer.
2.02 rounds to 2 and 5.01 rounds to 5. Thus, the estimate is 2×5 = 10.
The actual answer has 4 decimal places and is 10.1202 .
5.01
× 2.02
1002
100200
If you were to obtain an answer significantly different
from the estimate, you would know that an error
occurred.
10.1202
Multiplying and Dividing by Powers of Ten
When a decimal is multiplied by a power of 10, the decimal point is moved to the right by
the number of places indicated by the power on 10. Also, when a decimal is divided by a
power of 10, the decimal point is moved to the left by the number of places indicated by the
power on 10. These procedures were outlined in Chapter One and are given again here.
MULTIPLICATION BY A POWER OF TEN
When multiplying by a power of ten, move the decimal point to the
right by the number of places indicated by the exponent. Note that
for whole numbers the decimal point must be inserted to the right of
the ones place.
15
DIVISION BY A POWER OF TEN
When dividing by a power of ten, move the decimal point to the left
by the number of places indicated by the exponent. Note that for
whole numbers, the decimal point must be inserted to the right of the
ones place.
Example:
Example 7
235 ÷ 107 = 235.0 ÷ 107 = 0.0000235
Divide 3.455 ÷ 106 .
3.455 ÷ 106 = 0.000003455
6 places
Example 8
Multiply 456.003 × 107 .
456.003 × 107 = 4560030000 = 4,560,030,000
7 places
Division by a power of 10 is mathematically equivalent to multiplication by the negative
of that power of 10. For example, 4.52 ÷ 105 = 4.52 × 10 5 .
MATH FACT
Division by a power of 10 is equal to multiplication by 10 to the
negative of that power.
Example:
Example 9
8.93 ÷ 106 = 8.93 × 10
6
Multiply 5.67 × 10 4 .
5.67 × 10
4
= 5.67 ÷ 104 = 0.000567
16
Scientific Notation
Scientific notation is a method of writing very large and very small numbers by using powers
of 10. For example, the number 430,000,000,000 may be written as 4.3 × 1011. Using
scientific notation provides a way to write extremely large and small numbers in a compact
form that can be input into a scientific calculator. The procedure for writing a number in
proper scientific notation is given here.
PROCEDURE TO WRITE A NUMBER IN SCIENTIFIC NOTATION
1.
Move the decimal point so that there is exactly one non-zero digit to
the left of the decimal point.
2.
Count how many places you moved the decimal point. This number
is the power you place on 10.
3.
Multiply by 10 to a positive power when representing numbers large
in value. Multiply by 10 to a negative power when representing
numbers small in value.
Example 10 Represent 545,000,000,000,000 by using scientific notation.
First, move the decimal point so that there is 1 non-zero digit to the left of the
decimal point. Count how many places the decimal point is moved.
545,000,000,000,000 = 5.45000000000000 × ?
14 places
Since the number of places = 14, the power on 10 is 14. Also, since this is
a number large in value, we multiply by a positive power of ten.
545,000,000,000,000 = 5.45 × 1014
Note that the zeroes at the far right of 5.45 were deleted.
17
Example 11 Represent the number 0.000000002356 with scientific notation.
Move the decimal point so that there is 1 non-zero digit to the left of the
decimal point. Count how many places the decimal point is moved.
0.000000002356 = 000000002.356 × ?
9 places
Since the number of places = 9, the power on 10 is 9. Also, since this is a
number small in value, we multiply by a negative power of ten.
0.000000002356 = 2.356 × 10
9
Note that the zeroes to the left of 2 were deleted.
Scientific Calculators and Scientific Notation
In order to make good use of scientific notation, it is essential to know how to input these
numbers into a scientific calculator and interpret the results. The following guidelines apply
to most scientific calculators.
SCIENTIFIC NOTATION AND YOUR CALCULATOR
A number such as 2.4 × 103 will be displayed as 2.4 E 03 or 2.4 0 3 .
A number such as 7.9 × 10 8 will be displayed as 7.9 E 08 or 7.9
08
.
Input 5.6 × 1011 on a scientific calculator as 5 . 6 Exp 1 1 .
Input 7.9 × 10 4 on a scientific calculator as 7 . 9
Exp 4 +/- .
Example 12 If you multiplied 50,000 × 70,000,000 on a scientific calculator, how would
the answer of 3.5 × 1012 be represented?
This would be shown as 3.5
12
or 3.5 E12 .
18
Dividing Decimals
When a decimal is divided by a whole number using the long division process, the division
is performed as if the divisor and the dividend were whole numbers. The decimal point in
the answer is located directly above the decimal point of the dividend.
1.37
5 6.85
5
18
15
35
35
0
In this example, the decimal point in 1.37 is directly above the decimal point
in 6.85 . The answer to this division could be estimated by performing the
division 7 ÷ 5 = 1 2 or 1 r 2. From this estimation, we can see why the
5
decimal point must be placed after 1 in the answer. The procedure for
dividing a decimal by a whole number is given here.
PROCEDURE FOR DIVIDING A DECIMAL BY A WHOLE NUMBER
1.
Perform the division as if the divisor and the dividend were whole
numbers. Add zeroes to the far right of the decimal dividend if
needed.
2.
The decimal point in the answer is located directly above the decimal
point of the dividend.
3.
Continue the long division process so that there is one decimal place
more than what is desired in the rounded answer. This extra place
will be used for rounding.
4.
Round off the answer to the desired decimal place.
Example 13 Perform the division 3.77 ÷ 3 and round the answer to the nearest hundredth.
1.256
3 3.770
3
7
6
17
15
20
18
2
Note that a zero placeholder was added to 3.77 and it was
written as 3.770 . This division was performed in the same
way as the division 3770 ÷ 3 = 1256. The decimal point in
the answer was placed directly above the decimal point in
3.770.
Rounding 1.256 to the nearest hundredth results in the final answer of 1.26
.
19
Example 14 Perform the division 0.09 ÷ 4 and round the result to the nearest thousandth.
0.0225
4 0.0900
8
10
8
20
20
0
Two zero placeholders are added to 0.09 and it is written as
0.0900 . This division is performed in the same way as the
division 900 ÷ 4 = 225. The decimal point in the answer is
placed directly above the decimal point in 0.0900 .
Rounding 0.0225 to the nearest thousandth results in the answer 0.023 .
Example 15 Perform the division 0.32 ÷ 0.7 and round the result to the nearest hundredth.
In this example, we are not dividing by a whole number. We can, however,
change the form of this problem by moving the decimal point one place to the
right in both the divisor and the dividend.
Since 0.32 ÷ 0.7
= 0.32 , we may multiply the numerator and the
0.7
denominator of this fraction by 10 to obtain 0.32 × 10 = 3.2 which is 3.2 ÷7.
0.7 × 10
7
Thus, we perform the equivalent division 3.2 ÷7 .
.457
7 3.200
28
40
35
50
49
1
Two zero placeholders are added to 3.2 and it is written as
3.200 . This division is performed in the same way as the
division 3200 ÷ 7 . The decimal point in the answer is placed
directly above the decimal point in 3.200 .
Rounding 0.457 to the nearest hundredth results in 0.46 .
In Example 15, the divisor was not a whole number. The division was rewritten as an
equivalent division by moving the decimal points the same number of places in both the
divisor and the dividend. This technique is used in a general procedure for dividing
decimals. The procedure is given on the following page.
20
PROCEDURE FOR DIVIDING ANY TWO NUMBERS
1.
If the dividend is a whole number, insert a decimal point and as many
zeroes after the decimal point as are needed.
2.
If the divisor is not a whole number, make it into a whole number by
moving the decimal point to the right in both the divisor and the
dividend the same number of places.
3.
Use the procedure for dividing a decimal by a whole number.
Example 16 Perform the division 0.03 ÷ 2.9 and round the answer to the nearest
thousandth.
First move the decimal points one place to the right in both the divisor and
the dividend.
2.9 0.03
= 29 0.3
Now, use the procedure for dividing a decimal by a whole number.
.0103
29 0.300
29
10
0
100
87
13
The final answer, rounded to the thousandths place, is 0.010 .
Example 17 Perform the division 4.1 ÷ 0.4 and round the result to the tenths place.
First move the decimal points one place to the right in both the divisor and
the dividend.
0.4 4.1
10.25
4 41.00
4
1
0
10
8
20
20
0
= 4 41
The rounded answer is 10.3 .
21
MATH FACT
Before dividing decimals, estimate your answer, if possible, by
rounding decimals to whole numbers. If the estimate is not close to
the actual answer, there is probably an error in the actual calculation.
Example 18 If we divide 89.397 ÷ 9.9 and obtain a result of 0.0903, how would an
estimate of this answer indicate that an error was made?
The estimate of this division is 90 ÷ 10 = 9. The correct result of dividing
89.397 ÷ 9.9 is 9.03 .
Order of Operations
Order of operations for decimals is the same as it is for whole numbers. The order of
operations is given again here.
ORDER OF OPERATIONS
1.
Perform all operations within parentheses first.
2.
Perform exponent operations before multiplying, dividing, adding or
subtracting.
3.
Divide and multiply from left to right in the expression.
4.
Subtract and add from left to right in the expression.
Example 19 Evaluate 4.2 + 3.1 ÷ 0.12 .
First, multiply 0.1 by itself.
4.2 + 3.1 ÷ 0.12 = 4.2 + 3.1 ÷ (0.1 × 0.1)
= 4.2 + 3.1 ÷ 0.01
= 4.2 + 310
= 314.2
22
Example 20 Evaluate
(3.2
1.12) 3.4 2.0 × 2.1
0.1 × 2.0
Note that the fraction bar implies division. This expression is equivalent to
(3.2
1.12)
3.4
2.0 × 2.1
÷
0.1 × 2.0 .
First square 1.1 within the parentheses , then add this result to 3.2 . Note that
1.12 = 1.1 × 1.1 = 1.21 and 1.21 + 3.2 = 4.41 .
(3.2
1.12)
= 4.41
3.4
3.4
2.0 × 2.1
2.0 × 2.1
÷
÷
0.1 × 2.0
0.1 × 2.0
Next, perform the multiplication operations. Then, add and subtract from left
to right. Finally, divide the two results.
4.41
3.4
2.0 × 2.1
= 4.41
3.4
4.2
=
1.01
4.2
÷
=
5.21
÷
= 5.21 ÷ 0.2
= 26.05
0.2
÷
0.2
÷
0.2
0.1 × 2.0
23
3.4 Converting Fractions into Decimals
Since a fraction bar may always be interpreted as a division sign, the procedure for
converting a fraction into a decimal consists of dividing the numerator by the denominator
and then rounding the result. This procedure is given here.
PROCEDURE TO CHANGE A FRACTION INTO A DECIMAL
1.
Divide the numerator by the denominator.
2.
Carry out the division one place beyond the digit that is to be
rounded.
3.
Round the result.
Example 1
Write the fraction 3 as a decimal rounded to the thousandths place.
7
3
7
=3÷7
.4285
7 3.0000
28
20
14
60
56
40
35
5
Example 2
The rounded result is 0.429
Write the mixed number 4 3 as a decimal rounded to the thousandths place.
13
Since 4 3 = 4 + 3 , convert the fraction 3 into a decimal rounded to the
13
13
thousandths, and add this decimal to 4.
.2307
13 3.0000
26
40
39
10
0
100
91
9
13
The rounded decimal is 0.231 .
4 + 0.231 = 4.231 .
24
Example 3
5
Add 3
by converting each fraction into a decimal rounded to the
70
11
thousandths place, and then add the decimals.
The two divisions are
.04285
70 3.00000
280
200
140
600
560
400
350
50
.4545
11 5.0000
44
60
55
50
44
60
55
50
The rounded results add to 0.043 + 0.455 = 0.498
Example 4
1
Add 1 1
and obtain a fraction result. Then add by converting each
5
2
10
fraction into a decimal, and then add the decimals. Compare your results.
1
5
1
2
1
10
=
2
10
5
10
1
10
= 8
10
The decimal equivalents of the fractions are
0.2
5 1.0
10
0
0.5
2 1.0
10
0
0.1
10 1.0
10
0
These decimals add up to 0.2 + 0.5 + 0.1 = 0.8 .
These results are equal since 0.8 = 8 .
10
MATH FACT
Fractions may be added or subtracted on a scientific calculator in
one step by using the +, , and ÷ keys.
Example:
3
7
7
11
1
113
is input as 3 ÷ 7 + 7 ÷ 11
1 ÷ 113 = .
The result shown on the calculator is 1.0560855074. . .
25
Example 5
Use a scientific calculator to add 4455
11,231
nearest thousandth.
61
117
and round the answer to the
This is entered as 4455 ÷ 11231 + 61 ÷ 117 = and the displayed result is
.918037452807 which rounds to 0.918 .
Example 6
Use a scientific calculator to add 29,000
879,003
the nearest thousandth.
3441
13,417
and round the answer to
This is entered as 29000 ÷ 879003 + 3441 ÷ 13417 = and the displayed result
is .289457601676 which rounds to 0.289 .
MATH REMINDER
A scientific calculator uses the correct order of operations. Most
non-scientific calculators do not. A scientific calculator will
perform the divisions in 1 ÷ 2 + 2 ÷ 5 before adding and return a
correct result of 0.9.
A non-scientific calculator will usually perform all operations from
left to right and will incorrectly calculate 1 ÷ 2 + 2 ÷ 5 as 0.5 .
Example 7
A gas and oil mixture for a lawn mower requires 1 of a gallon of oil for each
8
gallon of gas. If 1 of one gallon of gas is added to the mower, how many
4
ounces of oil must be added? Note that 1 gallon is equivalent to 128 ounces.
Since 1 of a gallon of oil is added to 1 gallon of gas, 1 × 1 gallons of oil are
8
4
8
added to 1 of a gallon of gas. This means that 1 of a gallon of oil is added.
4
1
32
1
32
32
of one gallon is equivalent to 1 of 128 ounces.
32
is converted into the decimal 0.03125 .
0.03125 × 128 ounces = 4 ounces of oil