3.
DECIMAL FRACTIONS
IMPORTANT FACTS AND FORMULAS
I. 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 =.01;
99/100=99
hundreths=.99;7/1000=7 thousandths=.007,etc
II.
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=1/4;2.008=2008/1000=251/125.
III. 1. Annexing
zeros to the extreme right of a decimal fraction does not change its value
Thus, 0.8 = 0.80 = 0.800, etc.
Thus, 0.8 = 0.80 = 0.800, etc.
2. If numerator and
denominator of a fraction contain the same number of decimal
places, then we remove the decimal sign.
places, then we remove the decimal sign.
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
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.
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.
2. 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.
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 = 0.0730 x 10000 = 730.
3.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.
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, 2x2x2 = 8. Sum of decimal places = (1
+ 2 + 3) = 6. .2 x .02 x .002 = .000008.
4.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.
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.
5. 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*100)/(0.11*100) = (0.066/11) = 0.006V
V. 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.
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
VI. 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 = 0.3333….= 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 =
53/59 ;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 (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) / 90 =
15/19 = 1/6;
____
0.2273 = (2273 – 22)/9900 = 2251/9900
VII. Some
Basic 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
SOLVED EXAMPLES
Ex. 1.
Convert the following into vulgar fraction:
(i) 0.75 (ii)
3.004 (iii) 0.0056
Sol. (i). 0.75 = 75/100 = 3/4 (ii)
3.004 = 3004/1000 = 751/250 (iii)
0.0056 = 56/10000 = 7/1250
Ex. 2.
Arrange the fractions 5/8, 7/12, 13/16, 16/29 and 3/4 in ascending order of
magnitude.
Sol. Converting each of the given fractions into decimal form, we get :
5/8 = 0.624, 7/12 = 0.8125, 16/29 =
0.5517, and 3/4 = 0.75
Now,
0.5517<0.5833<0.625<0.75<0.8125
\ 16/29
< 7/12 < 5/8 < 3/4 < 13/16
Ex. 3.
arrange the fractions 3/5, 4/7, 8/9, and 9/11 in their descending order.
Sol. Clearly, 3/5 = 0.6, 4/7 = 0.571, 8/9 = 0.88, 9/111 = 0.818.
Now, 0.88 > 0.818 > 0.6 >
0.571
\ 8/9
> 9/11 > 3/4 > 13/ 16
Ex. 4. Evaluate : (i) 6202.5 + 620.25 + 62.025 + 6.2025 + 0.62025
(ii) 5.064 + 3.98 + 0.7036 + 7.6 + 0.3 + 2
Sol. (i) 6202.5 (ii)
5.064
620.25 3.98
62.025 0.7036
6.2025 7.6
+ __ 0.62025 0.3
6891.59775 _2.0___
19.6476
Ex. 5.
Evaluate : (i) 31.004 – 17.2368 (ii)
13 – 5.1967
Sol. (i) 31.0040 (ii) 31.0000
– 17.2386 – _5.1967
13.7654
7.8033
Ex. 6. What value will replace the question mark in the following equations ?
(i)
5172.49 + 378.352 + ? =
9318.678
(ii)
? – 7328.96 + 5169.38
Sol. (i) Let 5172.49 + 378.352 + x = 9318.678
Then
, x = 9318.678 – (5172.49 + 378.352) = 9318.678 – 5550.842 = 3767.836
(ii) Let x – 7328.96 = 5169.38. Then, x = 5169.38
+ 7328.96 = 12498.34.
Ex. 7. Find the products: (i) 6.3204 * 100 (ii) 0.069 * 10000
Sol. (i) 6.3204 * 1000 = 632.04 (ii) 0.069 * 10000 = 0.0690 * 10000 = 690
Ex. 8. Find
the product:
(i) 2.61 * 1.3 (ii) 2.1693 * 1.4 (iii) 0.4 * 0.04 * 0.004 * 40
Sol. (i) 261 8 13 = 3393. Sum of
decimal places of given numbers = (2+1) = 3.
2.61 * 1.3 = 3.393.
(ii) 21693 * 14 = 303702. Sum of
decimal places = (4+1) = 5
2.1693 * 1.4 = 3.03702.
(iii) 4 * 4 * 4 * 40 = 2560. Sum of
decimal places = (1 + 2+ 3) = 6
0.4 * 0.04 * 0.004 * 40 = 0.002560.
Ex. 9. Given
that 268 * 74 = 19832, find the values of 2.68 * 0.74.
Sol. Sum of decimal places = (2 +
2) = 4
2.68 * 0.74 = 1.9832.
Ex. 10. Find
the quotient:
(i) 0.63 / 9 (ii) 0.0204 / 17 (iii)
3.1603 / 13
Sol. (i) 63 / 9 = 7.
Dividend contains 2 places decimal.
0.63 / 9 = 0.7.
(ii) 204 / 17 = 12. Dividend contains 4
places of decimal.
0.2040 / 17 = 0.0012.
(iii) 31603 / 13 = 2431. Dividend
contains 4 places of decimal.
3.1603 / 13 = 0.2431.
Ex. 11.
Evaluate :
(i) 35 + 0.07 (ii) 2.5 + 0.0005
(iii)
136.09 + 43.9
Sol. (i) 35/0.07 = ( 35*100) / (0.07*100) = (3500 / 7) = 500
(ii) 25/0.0005 = (25*10000) /
(0.0005*10000) = 25000 / 5 = 5000
(iii) 136.09/43.9 = (136.09*10) /
(43.9*10) = 1360.9 / 439 = 3.1
Ex. 12. What value will come in place of question mark in the following equation?
(i) 0.006 +? = 0.6 (ii) ? + 0.025 = 80
Sol. (i) Let 0.006 / x = 0.6,
Then, x = (0.006 / 0.6) = (0.006*10) / (0.6*10) = 0.06/6 = 0.01
(ii) Let x / 0.025 = 80, Then, x = 80 *
0.025 = 2
Ex. 13. If (1
/ 3.718) = 0.2689, Then find the value of (1 / 0.0003718).
Sol. (1 / 0.0003718 ) = ( 10000 /
3.718 ) = 10000 * (1 / 3.718) = 10000 *
0.2689 = 2689.
___ ______
Ex. 14. Express as vulgar fractions : (i) 0.37 (ii) 0.053 (iii) 3.142857
__
___
Sol. (i) 0.37 = 37 / 99 . (ii) 0.053 = 53 / 999
______ ______
(iii) 3.142857 = 3 + 0.142857 = 3 +
(142857 / 999999) = 3 (142857/999999)
_ __ _
Ex. 15. Express as vulgar fractions : (i) 0.17 (ii) 0.1254 (iii) 2.536
_
Sol. (i)
0.17 = (17 – 1)/90 = 16 / 90 = 8/ 45
__
(ii) 0.1254 = (1254 – 12 )/ 9900 = 1242 / 9900 = 69 / 550
(iii) 2.536 = 2 + 0.536 = 2 +
(536 – 53)/900 = 2 + (483/900) = 2 + (161/300) = 2 (161/300)
Ex. 16. Simplify: 0.05 * 0.05 * 0.05 + 0.04 * 0.04 * 0.04
0.05 * 0.05 – 0.05 * 0.04 + 0.04 *
0.04
Sol. Given
expression = (a3 + b3)
/ (a2 – ab + b2), where a = 0.05 , b = 0.04
= (a
+b ) = (0.05 +0.04 ) =0.09
