IMPORTANT
FACTS AND FORMULAE
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
