Decimal Fraction
Decimal Fraction
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 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
 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
 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
 Operations on Decimal Fractions:
 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.
 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.
 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
 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 Ã· 17). Now, 204 Ã· 17 = 12.
Dividend contains 4 places of decimal. So, 0.0204 Ã· 17 = 0.0012
 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
 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
 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 nonrepeating digits.
Thus, 0.16 = 16 – 1 = 15 = 1 ; 0.2273 = 2273 – 22 = 2251 . 90 90 6 9900 9900
 Some Basic Formulae:
 (a + b)(a – b) = (a^{2} – b^{2})
 (a + b)^{2} = (a^{2} + b^{2} + 2ab)
 (a – b)^{2} = (a^{2} + b^{2} – 2ab)
 (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2(ab + bc + ca)
 (a^{3} + b^{3}) = (a + b)(a^{2} – ab + b^{2})
 (a^{3} – b^{3}) = (a – b)(a^{2} + ab + b^{2})
 (a^{3} + b^{3} + c^{3} – 3abc) = (a + b + c)(a^{2} + b^{2} + c^{2} – ab – bc – ac)
 When a + b + c = 0, then a^{3} + b^{3} + c^{3} = 3abc.
1



View Answer
Answer: Option B
Explanation:
Given Expression =  a^{2} – b^{2}  =  (a + b)(a – b)  = (a + b) = (2.39 + 1.61) = 4. 
a – b  (a – b) 
2.
What decimal of an hour is a second ?  

View Answer
Answer: Option C
Explanation:
Required decimal =  1  =  1  = .00027 
60 x 60  3600 
3.



View Answer
Answer: Option A
Explanation:
Given expression 





4.



View Answer
Answer: Option B
Explanation:
Given expression =  (0.1)^{3} + (0.02)^{3}  =  1  = 0.125 
2^{3} [(0.1)^{3} + (0.02)^{3}]  8 
5.
If 2994 ÷ 14.5 = 172, then 29.94 ÷ 1.45 = ?  

View Answer
nswer: Option C
Explanation:
29.94  =  299.4 
1.45  14.5 
=  2994  x  1  [ Here, Substitute 172 in the place of 2994/14.5 ]  
14.5  10 
=  172 
10 
= 17.2
6.
When 0.232323….. is converted into a fraction, then the result is:  

View Answer
Answer: Option C
Explanation:
0.232323… = 0.23 =  23 
99 
7.



View Answer
Answer: Option C
Explanation:
Let  .009  = .01; Then x =  .009  =  .9  = .9 
x  .01  1 
8.
The expression (11.98 x 11.98 + 11.98 x x + 0.02 x 0.02) will be a perfect square for xequal to:  

View Answer
Answer: Option C
Explanation:
Given expression = (11.98)^{2} + (0.02)^{2} + 11.98 x x.
For the given expression to be a perfect square, we must have
11.98 x x = 2 x 11.98 x 0.02 or x = 0.04
9.



View Answer
Answer: Option D
Explanation:
Given expression 








= 2.50 
10.
3889 + 12.952 – ? = 3854.002  

View Answer
Answer: Option D
Explanation:
Let 3889 + 12.952 – x = 3854.002.
Then x = (3889 + 12.952) – 3854.002
= 3901.952 – 3854.002
= 47.95.
11



View Answer
Answer: Option A
Explanation:
144  =  14.4 
0.144  x 
144 x 1000  =  14.4  
144  x 
x =  14.4  = 0.0144 
1000 
12.
View Answer
Answer: Option B
Explanation:
Suppose commodity X will cost 40 paise more than Y after z years.
Then, (4.20 + 0.40z) – (6.30 + 0.15z) = 0.40
0.25z = 0.40 + 2.10
z =  2.50  =  250  = 10. 
0.25  25 
X will cost 40 paise more than Y 10 years after 2001 i.e., 2011.
13.



View Answer
Answer: Option C
Explanation:
3  = 0.75,  5  = 0.833,  1  = 0.5,  2  = 0.66,  4  = 0.8,  9  = 0.9. 
4  6  2  3  5  10 
Clearly, 0.8 lies between 0.75 and 0.833.
4  lies between  3  and  5  .  
5  4  6 
14.
The rational number for recurring decimal 0.125125…. is:  

View Answer
Answer: Option C
Explanation:
0.125125… = 0.125 =  125 
999 
15.
617 + 6.017 + 0.617 + 6.0017 = ?  

View Answer
Answer: Option C
Explanation:
617.00
6.017
0.617
+ 6.0017

629.6357
