21
1. Decimal Fractions: Fractions in which denominators are powers of 10 are known as decimal fractions. Thu s, 1 = 1 tenth = .1; 1 = 1 hundredth = .01; 1 0 10 0 99 = 99 hundredths = .99; 7 = 7 thousandths = .007, etc.; 10 0 100 0 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 = 1 ; 2.008 = 200 8 = 25 1 . 10 0 4 100 0 12 5 3. 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. Thu s, 1. 84 = 18 4 = 8 . 2. 99 29 9 1 3 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

# Decimal Fraction

Embed Size (px)

Citation preview

1. 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 10

099

= 99 hundredths = .99;  7

= 7 thousandths = .007, etc.;100 1000

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 =1;       2.008 =2008=251.100 4 1000 125

3. 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

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 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.

iii. 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

iv. 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

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=0.00066 x 100=0.066= .0060.11 0.11 x 100 11

5. 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 3,6and7in descending order.5 7 9Now

,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

6. 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 99

9

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 non-repeating digits.

Thus, 0.16 =

16 - 1=15=1;   0.2273 =2273 -

22 =2251.90 90 6 9900 9900

7. Some Basic Formulae:i. (a + b)(a - b) = (a2 + b2)

ii. (a + b)2 = (a2 + b2 + 2ab)iii. (a - b)2 = (a2 + b2 - 2ab)iv. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)v. (a3 + b3) = (a + b)(a2 - ab + b2)

vi. (a3 - b3) = (a - b)(a2 + ab + b2)vii. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)

viii. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

1.  Evaluate

:(2.39)2 - (1.61)2

2.39 - 1.61A. 2 B.4

C.6 D.8

Explanation:

Given Expression =|a|2 -| b2

=(|a |+| b)(a - b)= (a + b) = (2.39 + 1.61) = 4.|a - |b (a - b)2.

What decimal of an hour is a second ?

A. .0025 B..0256

C..00027 D..000126

Explanation:

Required decimal = 1 = 1 = .0002760 x 60 36003.

The value of

(0.96)3 - (0.1)3 is:(0.96)2 + 0.096 + (0.1)2

A. 0.86 B.0.95

C.0.97 D.1.06

Explanation:

Given expression= (0.96)3 - (0.1)3

(0.96)2 + (0.96 x 0.1) + (0.1)2

= a3 - b3

a2 + ab + b2

= (a - b)= (0.96 - 0.1)

= 0.86

4.

The value of0.1 x 0.1 x 0.1 + 0.02 x 0.02 x 0.02is:0.2 x 0.2 x 0.2 + 0.04 x 0.04 x 0.04

C

A

A. 0.0125 B.0.125

C.0.25 D.0.5

Explanation:

Given expression =

(0.1)3 + (0.02)3

=1= 0.12523 [(0.1)3 + (0.02)3] 85.

If 2994 ÷ 14.5 = 172, then 29.94 ÷ 1.45 = ?

A. 0.172 B.1.72

C.17.2 D.172

Explanation:

29.94=299.41.45 14.5

=

2994x

1

14.5 10

= 17210

= 17.2

6.

When 0.232323..... is converted into a fraction, then the result is:

A.

15 B.29

C.2399 D. 23

100

B

C

C

Explanation:

0.232323... = 0.23 = 2399

7.  .009 = .01?

A. .0009 B..09

C..9 D.9

Explanation:

Let

.009= .01;     Then x =.009=.9= .9x .01 18.

The expression (11.98 x 11.98 + 11.98 x x + 0.02 x 0.02) will be a perfect square for x equal to:

A. 0.02 B.0.2

C.0.04 D.0.4

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.  (0.1667)(0.8333)(0.3333)is approximately equal to:

C

C

(0.2222)(0.6667)(0.1250)A. 2 B.2.40

C.2.43 D.2.50

Explanation:

Given expression=(0.3333)x(0.1667)(0.8333)(0.2222) (0.6667)(0.1250)

=3333

x

1x56 6

2222 2x 1253 1000

= 3x1x5x3x 82 6 6 2

=52

= 2.5010.

3889 + 12.952 - ? = 3854.002

A. 47.095 B.47.752

C.47.932 D.47.95

Explanation:

Let 3889 + 12.952 - x = 3854.002.

Then x = (3889 + 12.952) - 3854.002

= 3901.952 - 3854.002

= 47.95.

11.

D

D

0.04 x 0.0162 is equal to:

A. 6.48 x 10-3 B.6.48 x 10-4

C.6.48 x 10-5 D.6.48 x 10-6

Explanation:

4 x 162 = 648. Sum of decimal places = 6.So, 0.04 x 0.0162 = 0.000648 = 6.48 x 10-4

12.  4.2 x 4.2 - 1.9 x 1.9is equal to:2.3 x 6.1A. 0.5 B.1.0

C.20 D.22

Explanation:

Given Expression =

(a2 - b2) =(a2 - b2)= 1.(a + b)(a - b) (a2 - b2)13.

If144

=14.4

, then the value of x is:0.144 x

A. 0.0144 B.1.44

C.14.4 D.144

Explanation:

144 =14.4

B

B

A

0.144 x

144 x 1000 =14.4

144 x

x = 14.4= 0.0144100014.

The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2001, the price of commodity X was Rs. 4.20 and that of Y was Rs. 6.30, in which year commodity X will cost 40 paise more than the commodity Y ?

A. 2010 B.2011

C.2012 D.2013

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= 10.0.2

5 25

X will cost 40 paise more than Y 10 years after 2001 i.e., 2011.

15.

Which of the following are in descending order of their value ?

A.

1,2,3,4,5,63 5 7 5 6 7 B.1,2,3,4,5,63 5 5 7 6 7

C.1,2,3,4,5,63 5 5 6 7 7 D.6,5,4,3,2,17 6 5 7 5 3

B

D

Explanation:

No answer description available for this question. Let us discuss.

16.

Which of the following fractions is greater than3 and less than

5?4 6A.

12 B.23

C.45 D. 910

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.

4lies between3and5.5 4 617.

The rational number for recurring decimal 0.125125.... is:

A.

63487 B.119

993

C.125999 D.None of these

Explanation:

0.125125... = 0.125 = 125999

18.

C

C

617 + 6.017 + 0.617 + 6.0017 = ?

A. 6.2963 B.62.965

C.629.6357 D.None of these

Explanation:

617.00 6.017 0.617 + 6.0017 -------- 629.6357 --------- 19.

The value of

489.1375 x 0.0483 x 1.956is closet to:0.0873 x 92.581 x 99.749A. 0.006 B.0.06

C.0.6 D.6

Explanation:

489.1375 x 0.0483 x 1.956 489 x 0.05 x 2

0.0873 x 92.581 x 99.749 0.09 x 93 x 100

= 4899 x 93 x 10

=

163 x

1

279 10

=0.5810

= 0.058 0.06.

20.

C

B

0.002 x 0.5 = ?

A. 0.0001 B.0.001

C.0.01 D.0.1

Explanation:

2 x 5 = 10.

Sum of decimal places = 4

0.002 x 0.5 = 0.001

21.

34.95 + 240.016 + 23.98 = ?

A. 298.0946 B.298.111

C.298.946 D.299.09

Explanation:

34.95 240.016 + 23.98 -------- 298.946 -------- 22.

Which of the following is equal to 3.14 x 106 ?

A. 314 B.3140

C.3140000 D.None of these

B

C

C

Explanation:

3.14 x 106 = 3.14 x 1000000 = 3140000.

23.

The least among the following is:

A. 0.2 B.1 ÷ 0.2

C.0.2 D.(0.2)2

Explanation:

1 ÷ 0.2 = 1 =10= 5;0.2 2

0.2 = 0.222...;

(0.2)2 = 0.04.

0.04 < 0.2 < 0.22....<5.

Since 0.04 is the least, so (0.2)2 is the least.

24.  5 x 1.6 - 2 x

1.4 = ?1.3

A. 0.4 B.1.2

C.1.4 D.4

Explanation:

D

D

Given Expression =8 - 2.8

=5.2

=

52 = 4.

1.3 1.3 13

25.

How many digits will be there to the right of the decimal point in the product of 95.75 and .02554 ?

A. 5 B.6

C.7 D.None of these

Explanation:

Sum of decimal places = 7.

Since the last digit to the extreme right will be zero (since 5 x 4 = 20), so there will be 6 significant digits to the right of the decimal point.

26.

The correct expression of 6.46 in the fractional form is:

A.

64699 B.64640

1000

C.640100 D.640

99

Explanation:

6.46   = 6 + 0.46   = 6 +46  =594 + 46  =640.99 99 9927.

The fraction 101 27 in decimal for is:100000A. .01027 B..10127

B

D

C.101.00027 D.101.000027

Explanation:

101 27 = 101 + 27 = 101 + .00027 = 101.00027100000 10000028.

0.0203 x 2.92= ?0.0073 x 14.5 x

0.7A. 0.8 B.1.45

C.2.40 D.3.25

Explanation:

0.0203 x 2.92=

203 x 292=

4= 0.80.0073 x 14.5 x

0.773 x 145 x

7 5

29.

4.036 divided by 0.04 gives :

A. 1.009 B.10.09

C.100.9 D.None of these

Explanation:

4.036=403.6= 100.90.04 430.

C

A

C

3.87 - 2.59 = ?

A. 1.20 B.1.2

C.1.27 D.1.28

Explanation:

3. 87 - 2.59 = (3 + 0.87) - (2 + 0.59)

= 3 +

87 - 2 +

5999 99

= 1 +

87-5999 99

= 1 +

2899

= 1.28.

D