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Arithmetic Operations on Real Numbers
Arithmetic operations performed on rational numbers
.
Two fractions (rational numbers)
a
b
and
c
d
can be
added
by the means of the formula:
=
+
a
b
c
d
+
a d
b c
b d
.
For example,
=
+
7
6
4
15
+
( )
7
.
(
)
15
( )
4
.
( )
6
( )
6
.
(
)
15
=
+
105
24
90
=
129
90
.
The numerator 129 and denominator 90 have the greatest common divisor of 3, so we can divide the numerator and denominator by 3 to
obtain
43
30
.
Hence
=
+
7
6
4
15
43
30
,
where the result is given in its "
lowest terms
", that is, with the numerator and denominator having no common factor.
Alternatively, one can look for the
least common multiple
of the two denominators 6 and 15 in the original two fractions. This is the
lowest number that both 6 and 15 divide into exactly.
Since 6 = 2
x
3 and 15 = 3
x
5, the least common multiple of 6 and 15 is
2
x
3
x
5 = 30. The two fractions can now both be written with the common denominator of 30. This is achieved by multiplying both
numerator and denominator of
7
6
by 5 to give
=
7
6
35
30
, and by multiplying both numerator and denominator of
4
15
by 2 to give
=
4
15
8
30
.
Hence
=
+
7
6
4
15
+
35
30
8
30
=
43
30
,
as before.
The calculation is often written as follows.
=
+
7
6
4
15
+
35
8
30
=
43
30
.
The first number 35 in the numerator of the middle fraction is obtained by dividing the denominator 6 of the first fraction
into the common
denominator 30 to give 5, and then multiplying this number by the numerator 7. The second number 8 is obtained by dividing the
denominator 15 of the second fraction
into the common denominator 30 to give 2, and then multiplying this number by the numerator 4.
This calculation can be checked, at least partially, by performing an approximate decimal calculation with a calculator.
7
6
~
1.166666667
and
4
15
~
0.2666666667,
so that
+
7
6
4
15
~
1.433333334.
A decimal approximation for
43
30
can also be obtained.
43
30
~
1.433333333.
Two fractions (rational numbers)
a
b
and
c
d
can be
multiplied
by the means of the formula:
a
b
x
=
c
d
a c
b d
.
For example,
6
35
x
=
7
9
42
315
.
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View Full DocumentThe fraction
42
315
can be reduced to its lowest terms as follows.
=
42
315
6
45
=
2
15
.
Hence
6
35
x
=
7
9
2
15
.
Another way to perform the computation is to write the product in the intermediate form
6
35
x
=
7
9
6
.
7
35
.
9
and look for common factors of the top and bottom in this form. 7 is a common factor of the top and bottom so the 7 at the top can be
divided into the 35 at the bottom to give
=
6
.
7
35
.
9
6
5
.
9
.
The top and bottom can also be divided by 3 (6 and 9 have the common factor 3). Hence
=
6
5
.
9
2
5
.
3
=
2
15
.
The complete calculation may written in the form
6
35
x
=
7
9
6
.
7
35
.
9
=
6
5
.
9
=
2
15
.
It is common practice to use "
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 Spring '08
 Uri
 Real Numbers, Fractions

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