1.
Decimal fractions: fractions in which
denominators are powers of 10 are known as decimal fractions.
Thus, 1/10= 1 tenth = 1; 1/100 = 1
hundredth = 0.1;
99/100 = 99 hundredths = 9; 7/1000= 7
thousandths = .007, etc.
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 = 25/100 = ¼; 2. 008=
2008/1000 = 251/125.
3.
I. annexing zeros to the extreme right of a
decimal fraction does not change its value thus, 0.8 = 0.80= 0.800, etc.
II. if numerator and denominator of a
fraction contain the same number of decimal places, then we remove the decimal sign.
Thus,
1.84/2.99 = 184/299 =8/13; .365/584 = 365/584= 5/8
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 b s power
of 10: shift the decimal point to the r right by as many places as is the power
of 10.
Thus, 5.9632 x 100= 596. 32;
0.073 x 1000 = 0. 030 x 1000 = 730.
III.
Multiplication of decimal fractions: multiply the
given numbers considering them
without the 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 .02 x 002).
Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 +2 + 3) =6.
IV.
Dividing a decimal fraction by a counting number:
divide the given number without considering the decimal point, by the given counting
umber. 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, 0.00066/0.11 = 0.00066 x 100 / 0.11x 100 = 0.066/11 = 006.
VI.
Comparison of fraction is 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.
VII.
Suppose, we have to arrange the fractions 3/5,
6/7 and 7/9 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.
VIII.
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,.
In 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, 1/3= 0.333……= 0.3; 22/7 -=
3.142857142857……= 3. 142857.
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 = 5/9; 0.53 = 0.067 = 67 / 999 ‘etc.
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.
e.g., 0.17333…..= 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 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 / 90 = 15 \ 90= 1/6; 0. 2273 = 2273 – 22/9900 =
2251 / 9900.
IX.
Some basic formulate:
1.
( a + b) ( a – b) = (a² - b²).
2.
(a + b) ² = (a² +b² + 2 ab).
3.
(a – b)² = ( a² + b² - 2 ab).
4.
(a +b +c)² = a² + b²+ c² +2 ( ab + b c + ca).
5.
(a² + b³) = (a+ b) (a² - ab + b²).
6.
(a² - b³) = (a - b) (a² + ab + b²).
7.
( a³ + b³ + c³ - 3 abc) = ( a + b + c) ( a² +
b² + c² - ab – bc – ac.).
8.
When a + b + c = 0, then, a³ +b³ + c³ = 3 abc.
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