Fractions – Uttar Pradesh PCS Exam Notes

Fractions

Any unit can be divided into any numbers of equal parts, one or more of this parts is called fraction of that unit. e.g. one-forth (1/4), one-third (1/3), three-seventh (3/7) etc.

The lower part indicates the number of equal parts into which the unit is divided, is called denominator. The upper part, which indicates the number of parts taken from the fraction is called the numerator. The numerator and the denominator of a fraction are called its terms.

A fraction is unity, when its numerator and denominator are equal.
A fraction is equal to zero if its numerator is zero.
The denominator of a fraction can never be zero.
The value of a fraction is not altered by multiplying or dividing the numerator and the denominator by the same number.e.g. 2/3 = 2/6 = 8/12 = (2/4)/(3/4)
When there is no common factor between numerator and denominator it is called in its lowest terms.e.g. 15/25 = 3/5
When a fraction is reduced to its lowest term, its numerator and denominator are prime to each other.
When the numerator and denominator are divided by its HCF, fraction reduces to its lowest term.

Proper fraction: A fraction in which numerator is less than the denominator. e.g. 1/4, 3/4, 11/12 etc.

Improper Fraction: A fraction in which numerator is equal to or more than the denominator. e.g. 5/4, 7/4, 13/12 etc.

Like fraction: Fractions in which denominators are same is called like fractions.

e.g. 1/12, 5/12, 7/12, 13/12 etc.

Unlike fraction: Fractions in which denominators are not same is called, unlike fractions.

e.g. 1/12, 5/7, 7/9 13/11 etc.

Compound Fraction: Fraction of a fraction is called a compound fraction.

e.g. 1/2 of 3/4 is a compound fraction.

Complex Fractions: Fractions in which numerator or denominator or both are fractions, are called complex fractions.

Continued fraction: Fraction that contain additional fraction is called continued fraction.

e.g.

Rule: To simplify a continued fraction, begin from the bottom and move upwards.

Decimal Fractions: Fractions in which denominators are 10 or multiples of 10 is called, decimal fractions. e.g. 1/10, 3/100, 2221/10000 etc.

Recurring Decimal: If in a decimal fraction a digit or a set of digits is repeated continuously, then such a number is called a recurring decimal. It is expressed by putting a dot or bar over the digits. e.g.

Pure recurring decimal: A decimal fraction in which all the figures after the decimal point is repeated is called a pure recurring decimal.

Mixed recurring decimal: A decimal fraction in which only some of the figures after the decimal point is repeated is called a mixed recurring decimal.

Conversion of recurring decimal into proper fraction:

CASE I: Pure recurring decimal

Write the repeated digit only once in the numerator and put as many nines as in the denominator as the number of repeating figures. e.g.

CASE II: Mixed recurring decimal

In the numerator, take the difference between the number formed by all the digits after the decimal point and that formed by the digits which are not repeated. In the denominator, take the number formed as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits. e.g.

Questions

Level-I

Evaluate :	(2.39)² – (1.61)²
	2.39 – 1.61

A.	2
B.	4
C.	6
D.	8

What decimal of an hour is a second ?

A.	.0025
B.	.0256
C.	.00027
D.	.000126

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

The value of	0.1 x 0.1 x 0.1 + 0.02 x 0.02 x 0.02	is:
	0.2 x 0.2 x 0.2 + 0.04 x 0.04 x 0.04

A.	0.0125
B.	0.125
C.	0.25
D.	0.5

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

A.	0.172
B.	1.72
C.	17.2
D.	172

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

100

.009	= .01
?

A.	.0009
B.	.09
C.	.9
D.	9

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

(0.1667)(0.8333)(0.3333)	is approximately equal to:
(0.2222)(0.6667)(0.1250)

A.	2
B.	2.40
C.	2.43
D.	2.50

10.

3889 + 12.952 – ? = 3854.002

A.	47.095
B.	47.752
C.	47.932
D.	47.95

11.

Level-II

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

12.

4.2 x 4.2 – 1.9 x 1.9	is equal to:
2.3 x 6.1

A.	0.5
B.	1.0
C.	20
D.	22

13.

If	144	=	14.4	, then the value of x is:
	0.144		x

A.	0.0144
B.	1.44
C.	14.4
D.	144

14.

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

15.

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

1	,	2	,	3	,	4	,	5	,	6
3	,	5	,	7	,	5	,	6	,	7

1	,	2	,	3	,	4	,	5	,	6
3	,	5	,	5	,	7	,	6	,	7

1	,	2	,	3	,	4	,	5	,	6
3	,	5	,	5	,	6	,	7	,	7

6	,	5	,	4	,	3	,	2	,	1
7	,	6	,	5	,	7	,	5	,	3

16.

Which of the following fractions is greater than	3	and less than	5	?
	4		6

17.

The rational number for recurring decimal 0.125125…. is:

487

119

993

125

999

None of these

18.

617 + 6.017 + 0.617 + 6.0017 = ?

A.	6.2963
B.	62.965
C.	629.6357
D.	None of these

19.

The value of	489.1375 x 0.0483 x 1.956	is closest to:
	0.0873 x 92.581 x 99.749

A.	0.006
B.	0.06
C.	0.6
D.	6

20.

0.002 x 0.5 = ?

A.	0.0001
B.	0.001
C.	0.01
D.	0.1

Answers

Level-I

Answer:1 Option B

Explanation:

Given Expression =	a² – b²	=	(a + b)(a – b)	= (a + b) = (2.39 + 1.61) = 4.
	a – b		(a – b)

Answer:2 Option C

Explanation:

Required decimal =	1	=	1	= .00027
	60 x 60		3600

Answer:3 Option A

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

Answer:4 Option B

Explanation:

Given expression =	(0.1)³ + (0.02)³	=	1	= 0.125
	2³ [(0.1)³ + (0.02)³]		8

Answer:5 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

Answer:6 Option C

Explanation:

0.232323… = 0.23 =	23
	99

Answer:7 Option C

Explanation:

Let	.009	= .01; Then x =	.009	=	.9	= .9
	x		.01		1

Answer:8 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

Answer:9 Option D

Explanation:

Given expression

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

3333

1	x	5
6	x	6

2222

2	x	125
3	x	1000

=		3	x	1	x	5	x	3	x 8
		2		6		6		2

=	5
	2

= 2.50

Answer:10 Option D

Explanation:

Let 3889 + 12.952 – x = 3854.002.

Then x = (3889 + 12.952) – 3854.002

= 3901.952 – 3854.002

= 47.95.

Level-II

Answer:11 Option B

Explanation:

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

Answer:12 Option B

Explanation:

Given Expression =	(a² – b²)	=	(a² – b²)	= 1.
	(a + b)(a – b)		(a² – b²)

Answer:13 Option A

Explanation:

144	=	14.4
0.144	=	x

	144 x 1000	=	14.4
	144		x

x =	14.4	= 0.0144
	1000

Answer:14 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.

Answer:15 Option D

Answer:16 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

Answer:17 Option C

Explanation:

0.125125… = 0.125 =	125
	999

Answer:18 Option C

Explanation:

617.00

6.017

0.617

+ 6.0017

——–

629.6357

———

Answer:19 Option B

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

=	489
	9 x 93 x 10

=	163	x	1
	279		10

=	0.58
	10

= 0.058 0.06.

Answer:20 Option B

Explanation:

2 x 5 = 10.

Sum of decimal places = 4

0.002 x 0.5 = 0.001