Download 1. FINAL

Fractions
A fraction is a part of the whole (object, thing, region). It forms the part of basic aptitude of a person to have
and idea of the parts of a population, group or territory. Civil servants must have a feel of ‘fractional’ thinking. eg,
5
, here ‘12’ is the number of equal part into which the whole has been divided, is called denominator and ‘5’ is
12
the number of equal parts which have been taken out, is called numerator.
Example1: Name the numerator of
Solution: Numerator of
Denominator of
3
5
.
and denominator of
13
7
3
is 3.
7
5
is 13.
13
Lowest Term of a Fr action
Dividing the numerator and denominator by the highest common element (or number) in them, we get the
fraction in its lowest form.
eg, To find the fraction
6
in lowest form Since ‘2’ is highest common element in numerator 6 and denominator
14
14 so dividing them by 2, we get
Equiv alent
3
6
. Which is the lowest form of
.
7
14
Frac tions
If numerator and denominator of any fraction are multiplied by the same number then all resulting fractions
are called equivalent fractions.
eg,
1 2 3 4
1
, , , all are equivalent fractions but
is the lowest form.
2 4 6 8
2
Example 2: Find the equivalent fractions of
2
having numerator 6.
5
Solution: We know that 2 × 3 = 6. This means we need to multiply both the numerator and denominator by
3 to get the equivalent fraction.
Hence, required equivalent fraction 
2 2 3 6


5 5  3 15
Additio n and Subtraction of Fractions
Here two cases arise as denominators of the fraction are same or not.
Case I: When denominators of the two fractions are same then we write denominator once and add (or
subtract) the numerators.
2 3
5

=
7 7
7
eg,
Case II: If denominators are different, we need to find a common denominator that both denominators will
divide into.
1 3

6 8
eg,
1
2 3
4
,
=
=
6
12 18
24
We can write,
3
6
9
=
=
8
16 24
13
1 3
4
9


=
=
24
6 8
24 24

Example 3: Calculate
Solution.

1 3

2 7
1 1 7 7
3 3 2 6
=
=
and =
=
2 2  7 14
7 7  2 14
1 3
7
6
1


=
=
2 7
14 14 14
Multipl ication and Division of Fr actions
To multiply fractions, the numerators are multiplied together and denominators are multiplied together.
eg,
1 3
1 3 3
1

=
=
=
6 8
6  8 48 16
In division of fraction, the numerator of first fraction is multiplied by the denominator of second fraction and
gives the numerators. Also denominator of first fraction is multiplied by the nemerator of second fraction and
gives the denominator.
eg,
1 3

becomes
16 8
1 8 8 4

= =
6 3 18 9
P ro pe r a n d Im pro per F ra c t io ns
The fractions in which the number in numerator is less than that of denominator, are called proper fractions.
Also if the number in numerator is greater than that of denominator, then the fractions are called improper
frations.
eg,
4
3
is an improper fraction while is a proper fraction.
3
4
Mix e d Nu m be rs
A mixed number is that, which contains both a whole number and a fraction.
eg, 4
7 1 5
,3 ,6 are mixed numbers.
12 4 8
Example 4: Which of the following are proper and improper fractions?
7
6
12
5
(b)
(c)
(d)
9
5
7
13
Solution. (a) and (d) are proper fractions as numerator is less than denominator.
Also, (b) and (c) are improper fractions as numerator is greater than denominator.
(a)
Illusration 5: 
Solution.
Are
7
5
and 4 mixed number?
13
6
7
5
is only a proper fraction as it does not contain any whole number, while 4 is a mixed number
13
6
as it contains ‘4’ as a whole number and
D ec im al
5
is a fraction.
6
F ra c t io ns
The fractions in which denominators has the power of 10 are called decimal fractions.
eg,
0.25 = nought point two five
=
0.1 = nought point one =
25 1
= = one quarter
100 4
1
= one-tenth.
10
For converting a decimal fraction into simple fraction, we write the numerator without point and in the
denominator, we write ‘1’ and put the number of zeros as many times as number of digits after the point
in the given decimal fraction
eg,
0.037 =
37
1257
,0.1257 =
1000
10000
Example 6: Convert the each of the following decimal fractions into simple fractions.
(a) 5.76
(b) 0.023
(c) 257.5
Solution.
(a) 5.76 =
576 288 144
23
=
=
(b) 0.023 =
100
50
25
1000
(c) 257.7 =
2575 515
=
10
2
Addit io n or Subt rac t ion o f D e cim a l F rac t ions
In the addition or subtraction of decimal fractions, we write the decimal fractions in such a way that all the
decimal points are in the same straight line then these numbers can be add or subtract in simple manner.
Example 7: Solve 0.68 + 0.062 + 0.20
Solution.
0.680
+0.062
+0.200
0.942
Multip lication of Decimal Fractions
To multiply by multiplies (powers) of 10 the decimal point is moved to the right by the respective number of
zero.
Example 8: 0.75 × 10 = ?
Solution. 0.75 × 10 = 7.5 (The decimal point is shifted to right by one place)
To multiply decimals by number other than 10. We ignore the decimal point and multiply them in simple
manner and at last put the points after the number of digits (from right) corresponding to the given problem.
Example 9: Multiply 8 and 10.24
Solution. First we multiply 8 and 1024
8 × 1024 = 8192
Now,
8 × 10.24 = 81.92
(We put decimal points after two digits from right as
in given question).
Example 10: 12.4 × 1.62
Solution. We know that
124 × 162 = 20088
12.4 × 1.62 = 20.088
Division of Decimal Fr actions
Division of decimal numbers is the reverse of the multiplication case ie, we move the decimal point to the left
while dividing by multiplies of 10.
Example 11: 25.75 ÷ 10 = ?
Solution. 25.75 ÷ 10 = 2.575
When a decimal number is divided by an integer, then at first divide the number ignoring the decimal point
and at last put the decimal after the number of digits (from right) according at in the given problem.
Example 12: Divide 0.0221 by 17
Solution. First we divide 221 by 17 ie, without decimal 221 ÷ 17 = 13
Now, we put decimal according as in the given problem
0.0221 ÷ 17 = 0.0013
If divisor and dividend both are decimal numbers then first we convert them in simple fraction by putting
number of zero in the denominator of both. Then divide by the above manner.
Example 13: Divide by 0.0256 by 0.016
Solution.
0.0256 256  1000 256 25.6
=
=
=
=1.6
0.016
16  10000 160
16
Example 14: Divide 70.5 by 0.25
Solution.
70.5 705 100 7050

=
= 282
=
0.25 25 10
25
To Find HCF and LCM of Decimal Fr actions
First we make the decimal digits of the given decimal numbers, same by putting some number of zero if
necessary. Then find HCF or LCM ignoring decimals. And at last put the decimal according as the given numbers.
Example 15: Determine HCF and LCM of 0.27, 1.8 and 0.036.
Solution. Given numbers are 0.27, 1.8 and 0.036.
or
0.270, 1.800 and 0.036.
These numbers without decimals are 270, 1800 and 36.
Now,
HCF of 270, 1800 and 36 = 18
HCF of 0.270, 1.800 and 0.036 = 0.018

LCM of 270, 1800 and 36 = 18 × 5 × 2 × 3 × 10 = 5400
18 270
1800
36

5 15
100
2
2 3
20
2
3
10
1
LCM of 0.270, 1.800 and 0.36 = 5.400 or 5.4
Terminati ng and Non-Termi nating Recurring
Decimals
If decimal expression of any fraction be terminated then fraction is called terminating.
as
5
= 0.3125
16
But if we take example 33 ÷ 26, then
26) 33 (1.2692307
26
70
52
180  A
156
240
234
60
52
80
78
200
182
180  B
In this division, we see that remainder at the stages A and B are same. In the continued process of division by
26, the digits 6, 9, 2, 3, 0, 7 in the quotient will repeat onwards.
33
= 1.2692307692307...
26

This process of division is non terminating. Therefore, such decimal expressions are called non terminating
repeating (recurring) decimals.
In repeating digit, we put (–) bar.
33
= 1.2692307
26
ie,
Example 16: Write the following fractions in decimal form and till that these are terminating or non
terminating recurring.
2
3
Solution.
(b)
(a)
4
5
(c)
(a)
2
= 0.6666.... = 0.6 non terminating recurring
3
(b)
4
= 0.8 terminating
5
(c)
3
= 0.272727... = 0.27 non terminating recurring
11
(d)
17
= 0.1888... 0.18 non terminating recurring
90
No n- Te rm in at ing,
Non- R e curring
3
11
(d)
17
90
De c ima ls
Every fraction can be put in the form of terminating or non-terminating recurring decimals ie, these decimal
p
numbers can be put in the form of q . These are called rational numbers. But some decimals numbers are there
that can’t be put in the form of
p
, these are non-terminating, non-recurring decimals. Also these are called
q
irrational numbers.
eg, 0.101001000100001...
To c o n ve rt n o n-t e rmina ting re c urring dec im als into sim ple fra c tio n
First write the non-terminating recurring decimal in bar notation. Then write the digit ‘a’ in the denominators
as many times as number of digits recurring in the numerator. Also don’t put decimal in the numerator.
Example 17 Convert the following in simple fraction
(i) 0.33333...
(ii) 0.181818...
Solution.
3 1
(i) 0.33333... = 0.3 = =
9 3
18 2
(ii) 0.181818... = 0.18 =
=
99 11
Mixed Recurring Decimals
A decimal fraction in which some digits are not repeated and some are repeated, called mixed recurring
decimal.
Ho w to C on ve rt Mix ed R ec urring D ec im al int o a Simple F ra ct io n?
First we subtract non repeated part from the number (without decimal) and put number 9 as many times as
number of recurring digits and also put the number ‘0’ as many times odd number of non-recurring digits.
Example 18: Convert the following in simple fraction
(i)
(ii) 3.0072
0.18
Solution. (i) 0.18 =
18  1 17
=
90
90
2
72
2
= 3
=3
(ii) 3.0072 = 3 + 0.0072 = 3 
275
9900
275
Example 19: Arrange
2 13 4
15
, , and
in ascending order,
3 15 5
16
Solution. We know that
2
= 0.67
3
13
= 0.86
15
4
= 0.80
5
15
= 0.94
16
Here, it is clear that 0.67 < 0.80 < 0.86 < 0.94

2 4 13 15



3 5 15 16
7 8 5
2
Example 20: Arrange , ,
and is descending order.
9 11 13
7
Solution. We know that
7
= 0.77
9
8
= 0.72
11
5
= 0.38
13
2
= 0.28
7

Here it is clear that 0.77  0.72  0.38  0.28
7 8
5 2



9 11 13 7
EXERCISE
1. If
2025 = 45, then the value of
0.00002025 
0.002025 
20.25 =
(a) 49.95
(c) 4.9995
2. If
2
3
1
7
of 3  5 of
is equal to:
9
11
7
9
(a) 8
(b) 9
(c) 7/9
(d) 5/4
11. 9  1
2025 
(b) 49.5495
(d) 499.95
15 = 3.88, the the value of
12.
5
is:
3
(a) 1.39
(b) 1.29
(c) 1.89
(d) 1.63
3. If 2805 ÷ 2.55 = 1100, then 280.5 ÷ 25.5 is:
(a) 111
(b) 1.1
(c) 0.11
(d) 11
4. The value of 213 + 2.013 + 0.213 + 2.0013 is:
(a) 217.2273
(b) 21.8893
(c) 217.32
(d) 3.217.32
0.05  0.05  0.05  0.04  0.04  0.04
 ?
0.05  0.05  0.05  0.04  0.04  0.04
(a) 0.09
(b) 0.9
(c) 0.009
(d) 0.001
6. The numerator of a non-zero rational number is
five less than the denominator. If the denominator
is increased by eight and the numerator is
doubled, then again we get the same rational
number. The required rational number is:
(a) 1/8
(b) 4/9
(c) 2/8
(d) 3/8
5.
7. The simplification of 0.6  0.7  0.8  0.13 yields:
(d) 1.46
(b) 2.46
(c) 3.46
(a) 2.46
8. What will be the approximate value of 779.5 × 5
– 46.5 × 19 – 9?
(a) 10800
(b) 18008
(c) 10080
(d) 10008
9. A fraction becomes 2 when 1 is added to both the
numerator and the denominator and it becomes
3 when 1 is subtracted from both numerator and
denominator. The numerator of the given fraction
is:
(a) 7
(b) 5
(c) 3
(d) 8
10. The simplification of 3.36  2.05  1.33 equals:
(a) 2.46
(b) 2.61
(c) 2.64
(d) 2.64
1
1
1
1
1
1
1
1







20 30 42 56 72 90 110 132
is equal to:
(a) 3/6
(b) 1/7
(c) 1/6
(d) 1/10
13. The value of (0.3467  0.1333) is
(a) 4.38
(b) 0.4801
(c) 0.48
(d) 4.481
3
7  1 1
5 3
3
14. Simplify: 4 
of
    

5 8  3 4
7 4
7
1
6
2
(a) 2/9
(b) 56/77
(c) 50/73
(d) 3
2
9
15. What approximate value should come in place of
z in the following question?
242 + 2.42 + 0.242 + 0.0242 = z
(a) 670
(b) 575
(c) 580
(d) 680
16. What should come in place of the question mark
(?) in the following question?
248.8 – 190.48 + 68.08 = ?
(a) 126.4
(b) 127.4
(c) 128.4
(d) 124.6
17. Find a fraction such that if 1 is added to the
denominator it reduces to ½ and reduces to 3/5
on adding 2 to the numerator.
(a) 12/25
(b) 13/25
(c) 11/13
(d) 11/25
5.32  56  5.32  44
is:
7.662  2.342
(a) 14
(b) 13
(c) 10
(d) 12
19. The rational number, which equals the number
18. The value of
2.357 with recurring decimal is:
(a) 235/101
(b) 235/997
(c) 2335/1001
(d) 2379/997
© 2011 www.upscportal.com
24
Fractions
20. xy is a number that is divided by ab where xy <
ab and gives a result 0.xyxyxy... then ab equals:
(a) 77
(b) 33
(c) 99
(d) 66
21. Express
(a) 6
20
as mixed fraction.
3
2
3
22. Express 7
30
(a)
4
(b) 5
2
3
(c) 4
2
3
(d) 3
33
(c)
4
2
(b)
5
2
(c)
6
(a)
18
27
(b)
180
5
and
360
8
(c)
180
2
and
200
5
(d)
220
2
and
550
5
(a)
34
(d)
4
1
2
(b)
3
2
(d) 9
6025
241
(b)
10000
400
29. Solve
(a)
2
(d)
7
to
25. Simplest form of
(b)
(c)
2
3
(d)
4
3
(d)
605
844
(d)
5
8
28. Simplest fraction form of 0.6025 is
24. The numerator of the fraction which is equivalent
15
with denominator 7 is
35
(a) 5
(b) 3
(c) 7
250
2
and
400
3
(a) 1
1
23. Which one of the following is equivalent to ?
3
1
(a)
2
(a)
27. Which of the following is proper fraction?
2
3
3
as improper fraction.
4
31
(b)
4
26. Which of the following is proper fraction?
30.
(c)
525
700
1.15
to its simplest form.
1.84
115
184
(b)
23
184
(c)
115
23
Solve.
0.635 + 1.8 + 2.9 + 18.358 + 0.02345
(a) 23.78645
(b) 22.87654
(c) 23.88456
(d) 24.78645
36
is
54
6
9
(c)
2
3
(d)
1
3
ANSWERS
1. (b)
11. (a)
21. (a)
2. (b)
12. (c)
22. (b)
3. (d)
13. (b)
23. (c)
4. (a)
14. (d)
24. (b)
5. (a)
15. (d)
25. (c)
6. (d)
16. (a)
26. (d)
7. (a)
17. (b)
27. (c)
8. (a)
18. (c)
28. (b)
9. (a)
19. (d)
29. (d)
10. (d)
20. (c)
30. (a)
EXPLANATIONS
1.
2025
= 45
= 3  2  1
0.00002025  0.002025 
= 0.0045 + 0.045 + 45 + 4.5
= 49.5495
2025 
20.25
11.
5
=
3
3.
1100 
1
100
= 11
 0.05 3   0.04 3
 0.05 2  0.05  0.04   0.04 2
5.
if 0.05 = a, 0.04 = b
a3  b3
=a+b
a   ab  b2
= 0.05 + 0.04 = 0.09
6. Let the denominator be x, then by the given
x 5
. Also,
condition the rational number is
x
2  x  5
x 5
2
1
=
=
or
or 2x = x + 8 or x = 8
x 8
x
x
x 8
then,

 11 36   36 7 

  =9–4÷4
  
 9 11   7 9 
=9–1=8
9 
5 3

=
3 3
280.5 ÷ 25.5
3.88
15
= 1.29
=
3
3
2805
=
2.55  10  10
The rational number is
3
.
8
7. 0.6  0.7  0.8  0.13
6 7 8 13
246



=
= 2.46
9 9 9 99
99
8. 779.5 × 15 – 46.5 × 19 – 9 = ?
= 11692.5 – 883.5 – 9
= 11692.5 – 892.5 = 10800
=
10. 3.36  2.05  1.33
2
3
1
5
of 3  5 of
7
9
11
7
Using BODMAS rule,
15 = 3.88
2.
9 1
36 5 33
64


= 2
= 2.64
99 99 99
99
3467  34
3433
,
=
9900
9900
1333  13
1320
=
0.1333 =
9900
9900
3433 1320

 .03467  0.1333 =
9900 9900
4753
=
= 0.4801.
9900
3
2
4  7   1  1   5  3 of 3
14.
5 8  3 4  7 4
7
1
6
Using BODMAS rule,
13.
0.3467 =
11
4 
=
11
6
 6
= 
 4
 6
= 
 4
 12

=
 7
7 7 5  3 3


   
8 12 7  4 7 
7
7  5 9 
  
 

8  12  7 28 
8
7  5 28 
  
 

7  12  7 9 
7  20
20
= 1
 
12 
9
9
29
2
= 3 .
=
9
9
357
999
2  299  357
2355
=
=
999
999
xy
.
20. 0.xy =
99
21. 3) 20 (6
18
2
19.
2.357
= 2  0.357 = 2 
20
2
= 6
3
3

22. 7
3 31
=
4 4
23.
1 1 2 2
=
=
3 3 3 6
24.
15
15  5
2
=
=
35
35  5
3
25.
36 36  18 2
=
=
54 54  18 3
26.
220
22 2
=
=
550
55 5
27.
2
is proper fraction as 2 is less than 3.
3
28. 0.6025 =
6025
241
=
10000 400
1.15
115
115  23
5
=
=
=
1.84
184
184  23
8
30.
0.635
1.87
2.9
18.358
+ 0.02345
23.78645
29.