DECIMAL FRACTIONS
In the study of whole numbers, we introduced the concept of place value. This allowed us to
represent a large number of objects in a compact way by giving a different meaning to a digit
depending upon its location relative to the units place. Thus the number 30 means thirty objects,
whereas the number 300 means three hundred objects. This concept can be extended to indicate
fractional numbers of objects, and requires the introduction of the decimal point. The decimal
point separates the units place from the fractional part of a number. The place immediately to
the right of the decimal point is called the tenths place, and the digit located there indicates the
number of tenths of an object. The place to the right of the tenths place is the hundredths place,
and so on.
7 3 6 1 . 2 9
8
Place title
Place Value
Thousandths
1/1,000
Hundredths
1/100
Tenths
1/10
Units
1
Tens
10
Hundreds
100
Thousands
1,000
Figure 3-1 Decimal Place Relationships
Numbers with digits to the right of the decimal place are decimal fractions. Numbers consisting
of a whole number and a decimal fraction are sometimes called mixed decimals. The magnitude
of a decimal fraction or a mixed decimal is the sum of the digits each multiplied by its place
value. The procedure is exactly the same as computing the magnitude of a whole number, as
was done in Lesson 1.
The magnitude of 7,361.298 is:
Digit Place Value
7 ×
1,000
7,000
=
3 ×
100
=
6
1 ×
×
=
300
10
60
1
=
1
2 ×
1/10
200/1,000
=
2/10 =
9 ×
1/100
=
9/100 = 90/1,000
8 ×
1/1,000 =
8/1,000 = 8/1,000
Sum =
7,361
= 7,361
=
+
298
1,000
298
1,000
7,361.298
Example 3-1
Notice in the example above that we found the LCD of the fractional parts so that they could be
added directly.
A common fraction can be changed to a decimal fraction by performing the division that is
indicated by the common fraction.
Find the decimal equivalent of
3
.
4
3
= 3÷4 =
4
0.75
4 3.00
28
20
20
0
3
= 0.75
4
Example 3-2
75
75
. Reducing
to its lowest terms, by dividing numerator
100
100
3
and denominator by 25 results in the fraction .
4
The number 0.75 is the same as
There are some common fractions that do not have exactly equivalent decimal fractions.
The decimal equivalent of
2
is:
3
Example 3-3
Rounding normally approximates this type of decimal fraction, which will be studied later.
In a previous example, we found the decimal equivalent of a common fraction by dividing. We
can also do the reverse process, changing a decimal fraction to a common fraction, by writing the
decimal as a common fraction and then reducing it to lowest terms.
Write the common fraction that is equivalent to
the decimal fraction 0.375.
0.375 =
375
375 ÷ 125 3
=
=
1,000 1,000 ÷ 125 8
Recall from the previous chapter to reduce the
fraction to lowest terms, factor each term into
its smallest component.
375
3× 5× 5× 5
=
1,000 2 × 2 × 2 × 5 × 5 × 5
Then cancel out each factor that occurs in each
term.
3x5x5x5
2x2x2x5x5x5
Then multiply the factors in each term
together.
3
3
=
2x2x2
8
Thus simplified 0.375 in a fractional form is
3
.
8
Example 3-4
Remember that the decimal fraction 0.375 is just a compact way of writing:
(3 ×
1
1
1
) + (7 ×
) + (5 ×
)
10
100
1,000
=
3
7
5
300
70
5
+
+
=
+
+
10 100 1,000 1,000 1,000 1,000
=
375
1,000
ADDITION AND SUBTRACTION OF DECIMALS
The addition and subtraction of decimals is accomplished in exactly the same manner as the
addition and subtraction of whole numbers. The numbers are placed in a vertical column, taking
care that the place values are aligned. In the case of decimals, aligning the decimal points can
ensure this.
Find the sum of 39.62, 41.093, and 0.0327.
39.62
41.093
+ 0.0327
80.7457
Example 3-5
Notice that the principle of “carrying” from one place to the next is also utilized in adding
decimals.
Subtraction is done in the same manner as with whole numbers.
32.100
–16.379
15.721
Example 3-6
There are two points to notice in this example. First, it is sometimes necessary to add zeros to
the right of a decimal fraction as an aid in the subtraction process. This does not change the
value of the decimal; that is, 32.1 is exactly equal to 32.100. Second, we have utilized the
principle of “borrowing” from one place and adding to another.
MULTIPLICATION OF DECIMALS
The multiplication of decimals can be accomplished in two ways. In the first method, the
numbers are placed one above the other and the multiplication carried out without regard to the
decimal points, just as with whole numbers. The decimal point in the product is located by
adding the number of digits to the right of the decimal points in the multiplicand and the
multiplier and placing the decimal point in the product with this number of digits to the right of
the decimal point.
Multiply 16.2 and 1.15. Multiply without
concern for the decimal.
162
×
115
810
162
162
18630
Example 3-7
Then place the decimal in your answer based on the number of decimal places in the
multiplicand and multiplier.
16.2 multiplicand has 1 decimal place
1.15 multiplier has 2 decimal places
Therefore the product must have 3 decimal places:
18 630
Note it is important to keep track of all zeroes until the decimal is placed. Thus the answer is
18.630. However, since the zero is on the end it may be omitted unless the need for significant
digits is required. (Topic on significant digits to be discussed later).
Therefore 18.63 is an acceptable answer.
Converting the decimals to improper fractions and multiplying directly could also carry out the
multiplication.
Multiply 16.2 and 1.15.
16.2 = 16
1.15 = 1
2 162
=
10 10
15 115
=
100 100
162 115 18,630
×
=
= 18.630
10 100 1,000
Example 3-8
DIVISION OF DECIMALS
In dividing decimals, the decimal point in the divisor is moved all the way to the right and the
decimal point in the dividend is moved the same number of places to the right. Division is then
carried out and the decimal point in the quotient is located directly above the decimal point in the
dividend.
Divide 41.05 by 2.5. Divide without concern
for the decimal.
1642
25 4105
25
160
150
105
100
50
50
0
Example 3-9
Move the decimal in the divisor all the way to the right. Count the number of places it is moved.
one place.
2.5
Move the decimal in the 41.05 dividend the same number of places to the right: one
place.
16.42
2.5 41.05
Place the decimal in the quotient directly above the decimal in the dividend.
Notice that the process of moving the decimal point is equivalent to multiplying the dividend and
divisor by factors of 10 so that the divisor becomes a whole number. Converting the decimals to
improper fractions and then dividing can also carry out the division.
Divide 41.05 by 2.5.
5
4,105
41.05 = 41
=
100
100
2.5 = 2
5 25
=
10 10
4,105
100 = 4,105 × 10 = 4,1050
25
100 25
2500
10
4,1050
2 × 5 × 5 × 821
=
2500
2× 2×5× 5× 5×5
=
1,642
100
= 16.42
Example 3-10