1. Decimal
Fractions:
Fractions in which
denominators are powers of 10 are known as decimal
fractions.
Thus,
|
1
|
= 1 tenth = .1;
|
1
|
= 1 hundredth = .01;
|
10
|
100
|
99
|
= 99 hundredths = .99;
|
7
|
= 7 thousandths = .007, etc.;
|
100
|
1000
|
2. Conversion
of a Decimal into Vulgar Fraction:
Put 1 in the denominator
under the decimal point and annex with it as many zeros as is the number of
digits after the decimal point. Now, remove the decimal point and reduce the fraction
to its lowest terms.
Thus, 0.25 =
|
25
|
=
|
1
|
; 2.008 =
|
2008
|
=
|
251
|
.
|
100
|
4
|
1000
|
125
|
3. Annexing
Zeros and Removing Decimal Signs:
Annexing zeros to the
extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80
= 0.800, etc.
If numerator and
denominator of a fraction contain the same number of decimal places, then we
remove the decimal sign.
Thus,
|
1.84
|
=
|
184
|
=
|
8
|
.
|
2.99
|
299
|
13
|
4. Operations
on Decimal Fractions:
i.
Addition and Subtraction of Decimal Fractions: The
given numbers are so placed under each other that the decimal points lie in one
column. The numbers so arranged can now be added or subtracted in the usual
way.
ii.
Multiplication of a Decimal Fraction By a
Power of 10: Shift the decimal point to the right by as many places as is
the power of 10.
Thus, 5.9632 x 100 =
596.32; 0.073 x 10000 = 730.
iii.
Multiplication of Decimal Fractions: Multiply
the given numbers considering them without decimal point. Now, in the product,
the decimal point is marked off to obtain as many places of decimal as is the
sum of the number of decimal places in the given numbers.
Suppose we have to find the
product (.2 x 0.02 x .002).
Now, 2 x 2 x 2 = 8. Sum of
decimal places = (1 + 2 + 3) = 6.
.2
x .02 x .002 = .000008
iv.
Dividing a Decimal Fraction By a Counting
Number: Divide the given number without considering the decimal
point, by the given counting number. Now, in the quotient, put the decimal
point to give as many places of decimal as there are in the dividend.
Suppose we have to find the
quotient (0.0204 ÷ 17). Now, 204 ÷ 17 = 12.
Dividend contains 4 places
of decimal. So, 0.0204 ÷ 17 = 0.0012
v.
Dividing a Decimal Fraction By a Decimal
Fraction: Multiply both the dividend and the divisor by a suitable
power of 10 to make divisor a whole number.
Now, proceed as above.
Thus,
|
0.00066
|
=
|
0.00066 x 100
|
=
|
0.066
|
= .006
|
0.11
|
0.11 x 100
|
11
|
5. Comparison
of Fractions:
Suppose some fractions are
to be arranged in ascending or descending order of magnitude, then convert each
one of the given fractions in the decimal form, and arrange them accordingly.
Let us to arrange the fractions
|
3
|
,
|
6
|
and
|
7
|
in descending order.
|
5
|
7
|
9
|
Now,
|
3
|
= 0.6,
|
6
|
= 0.857,
|
7
|
= 0.777...
|
5
|
7
|
9
|
Since, 0.857 > 0.777... > 0.6.
So,
|
6
|
>
|
7
|
>
|
3
|
.
|
7
|
9
|
5
|
6. Recurring
Decimal:
If in a decimal fraction, a
figure or a set of figures is repeated continuously, then such a number is
called a recurring decimal.
n a recurring decimal, if a
single figure is repeated, then it is expressed by putting a dot on it. If a
set of figures is repeated, it is expressed by putting a bar on the set.
Thus,
|
1
|
= 0.333... = 0.3;
|
22
|
= 3.142857142857.... = 3.142857.
|
3
|
7
|
Pure
Recurring Decimal: A decimal fraction, in which all the figures after the
decimal point are repeated, is called a pure recurring decimal.
Converting
a Pure Recurring Decimal into Vulgar Fraction: Write
the repeated figures only once in the numerator and take as many nines in the
denominator as is the number of repeating figures.
Thus, 0.5 =
|
5
|
; 0.53 =
|
53
|
; 0.067 =
|
67
|
, etc.
|
9
|
99
|
999
|
Mixed
Recurring Decimal: A decimal fraction in which some figures do not repeat and
some of them are repeated, is called a mixed recurring decimal.
Eg. 0.1733333.. = 0.173.
Converting
a Mixed Recurring Decimal Into Vulgar Fraction: In
the numerator, take the difference between the number formed by all the digits
after decimal point (taking repeated digits only once) and that formed by the
digits which are not repeated. In the denominator, take the number formed by as
many nines as there are repeating digits followed by as many zeros as is the
number of non-repeating digits.
Thus, 0.16 =
|
16 - 1
|
=
|
15
|
=
|
1
|
; 0.2273 =
|
2273 - 22
|
=
|
2251
|
.
|
90
|
90
|
6
|
9900
|
9900
|
7. Some
Basic Formulae:
i.
(a + b)(a - b) = (a2 + b2)
ii.
(a + b)2 = (a2 +
b2 + 2ab)
iii.
(a - b)2 = (a2 +
b2 - 2ab)
iv.
(a + b + c)2 = a2 +
b2 + c2 + 2(ab + bc + ca)
v.
(a3 + b3) = (a + b)(a2 - ab + b2)
vi.
(a3 - b3) = (a - b)(a2 + ab + b2)
vii.
(a3 + b3 + c3 -
3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
viii.
When a + b + c =
0, then a3 + b3 + c3 =
3abc.
