# DECIMAL FRACTIONS (PART-I)

I. Decimal Fractions : Fractions during which denominators are powers of 10 are generally known as decimal  fractions.

Thus ,1/10=1 tenth=.1;1/100=1
hundredth =.01;
99/100=99
hundreths=.99;7/1000=7 thousandths=.007,and many others
II.  Conversion of a Decimal Into Vulgar Fraction : Put 1 within the denominator underneath the decimal level and annex
with it as many zeros as is the variety of digits after the decimal level. Now,
take away the decimal level and scale back the fraction to its lowest phrases.
Thus,
0.25=25/100=1/4;2.008=2008/1000=251/125.

III.  1.  Annexing
zeros to the acute proper of a decimal fraction doesn’t change its worth

Thus, 0.8 = 0.80 = 0.800,
and many others.
2.  If numerator and denominator of a fraction
comprise the identical variety of decimal locations, then we take away the decimal
signal.
Thus,
1.84/2.99 = 184/299 = 8/13;
0.365/0.584 = 365/584=5
IV.  Operations on Decimal Fractions :

1.  Addition and Subtraction of Decimal
Fractions
:
The
given numbers are so
positioned underneath one another that the decimal factors lie in a single column. The numbers so organized can now be added or subtracted within the common approach.

2.  Multiplication of a Decimal
Fraction By a Energy of 10
:
Shift the decimal level to the proper by as many locations as is the ability of 10.

Thus, 5.9632 x 100 = 596,32; 0.073 x 10000 = 0.0730 x 10000
= 730.

3. Multiplication
of Decimal Fractions
: Multiply the given numbers contemplating them with out the decimal level. Now, within the product, the decimal level is
marked
off to acquire as many locations of decimal as is the sum of the variety of decimal locations within the given numbers.

Suppose we have now to search out the
product (.2 x .02 x .002). Now, 2x2x2 = 8. Sum of decimal locations = (1 + 2 + 3)
= 6. .2 x .02 x .002 = .000008.

4. Dividing a
Decimal Fraction By a Counting Quantity :
Divide the given quantity with out contemplating the decimal level, by the given counting quantity. Now, within the quotient, put the decimal level to provide as many locations of decimal
as
there are within the dividend.

Suppose we have now to search out the
quotient (0.0204 + 17). Now, 204 ^ 17 = 12. Dividend comprises 4 locations of
decimal. So, 0.0204 + 17 = 0.0012.
5. Dividing a Decimal Fraction By a Decimal
Fraction :
Multiply each the dividend and the divisor by an appropriate energy
of 10 to make divisor an entire quantity. Now, proceed as above.

Thus,
0.00066/0.11 =
(0.00066*100)/(0.11*100) = (0.066/11) = 0.006V

V.  Comparability of Fractions : Suppose some fractions are to be organized in ascending or
descending order of magnitude. Then, convert every one of many given fractions in
the decimal kind, and organize them accordingly.
Suppose, we have now to rearrange the
fractions  3/5, 6/7 and seven/9  in descending order.
now, 3/5=0.6,6/7 = 0.857,7/9 =
0.777….
since  0.857>0.777…>0.6, so 6/7>7/9>3/5

VI. Recurring Decimal : If in a decimal fraction, a determine or a set of figures is
repeated repeatedly, then such a quantity is known as a recurring decimal.
In a recurring decimal, if a single determine is repeated,
then it’s expressed by placing a dot on it. If a set of figures is repeated,
it’s expressed by placing a bar on the set

Thus 1/3 = 0.3333….= 0.3;
22 /7 = 3.142857142857…..= 3.142857
Pure Recurring Decimal: A decimal fraction during which all of the figures after the
decimal level are repeated, is known as a pure recurring decimal.
Changing a Pure Recurring
Decimal Into Vulgar Fraction :
Write the
repeated figures solely as soon as within the numerator and take as many nines within the
denominator as is the variety of repeating figures.
thus
,0.5 = 5/9;  0.53 = 53/59  ;0.067 =
67/999;and many others…

Combined Recurring Decimal: A decimal fraction during which some figures don’t repeat and
a few of them are repeated, is known as a combined recurring decimal.
e.g., 0.17333 =
0.173.

Changing a Combined Recurring Decimal Into Vulgar Fraction : Within the numerator, take the distinction between the quantity
shaped by all of the digits after decimal level (taking repeated digits solely as soon as)
and that shaped by the digits which aren’t repeated, Within the denominator, take
the quantity shaped by as many nines as there are repeating digits adopted by as
many zeros as is the variety of non-repeating digits.
Thus 0.16 = (16-1) / 90 =  15/19 =
1/6;
____
0.2273 =  (2273 – 22)/9900 = 2251/9900
VII.  Some Primary
Formulae :
1.   (a +
b)(a- b) = (a2 – b2).
2.   (a +
b)2 = (a2 + b2 + 2ab).
3.
(a – b)2 = (a2 + b2
2ab).
4.    (a + b+c)2 = a2 + b2 + c2+2(ab+bc+ca)
5.    (a3
+ b3) = (a + b) (a2 – ab + b2)
6.    (a3
– b3) = (a – b) (a2 + ab + b2).
7.    (a3
+ b3 + c3 – 3abc) = (a + b + c) (a2 + b2
+ c2-ab-bc-ca)
8.
When   a + b + c = 0, then a3
+ b3+ c3 = 3abc