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SUBJECTABLE MATHMATICS
PREFACE
(A NOTE TO THE READER)
This book, A COMPLETE BOOK ON SUBJECTIVE (CONVENTIONAL) ARITHMETIC has been specially prepared for candidates
appearing for competitive entrance examination held by Staff Selection
Commission (S.S.C.).
Syllabus of ARITHMETIC
Questions are designed to test the ability of arithmetical
computations of whole numbers, decimals and fractions. The questions
are based on arithmetical concepts of various topics and the candidates
have to give a detailed explanation in order to arrive at the final result of
the problem.
Every topic of arithmetic in this book starts with the introduction
which is followed by plenty of solved examples with complete
explanation that can help the candidates to secure full marks.
At the end of every chapter, an excercise containing different types
of questions is given for your practice. All questions are fully solved with
proper explanation.
I am confident that this book will be very useful to the candidates.
Best of Luck.
S.L. Gulati
CONTENTS
Preface
1. Numbers and Simplifications
v
1
2. H.C.F and L.C.M.
35
3. Ratio and Proportion
49
4. Percentage
79
5. Average
100
6. Profit and Loss
114
7. Interest
137
8. Time and Work
161
9. Time and Distance
185
10. Mensuration
210
11. Data Interpretation
264
12. Partnership
281
13. Alligation
302
1
NUMBERS AND
SIMPLIFICATIONS
1. ARITHMETIC is a science that deals with numbers, and of the
methods of computing by means of numbers.
2. Integers: Numbers like –3, –2, –1, 0, 1, 2, 3 etc. are called integers.
3. Rational numbers: All numbers of the type p/q where q ¹ 0 and p
is an integer are called rational numbers; e.g. –2,
7
4
9
, – , 0, 2,
3
7
13
etc. are all rational numbers.
4. Irrational numbers: Numbers like 2 , 3 , 5 etc. are called
irrational numbers.
5. Real numbers: All rational numbers, irrational numbers and a
combination of rational and irrational numbers are called real
numbers; e.g. –5, 7/3, 2 , + 3 etc. real numbers.
6. Natural numbers: The numbers 1, 2, 3, 4,…are called natural
numbers.
(i) Sum of the first n natural numbers, i.e.
n ´ (n + 1)
2
(ii) Sum of the first n square numbers, i.e.
1+2+3+4+…+n=
12 + 22 + 32 + 42 + … + n2 =
n ( n + 1) ( 2 n + 1)
6
2
Subjective Arithmetic
(iii) Sum of the first n cube numbers i.e.
13 + 23 + 33 + 43 + … + n3 =
LM n (n + 1) OP
N 2 Q
2
Þ 13 + 23 + 33 + 43 + … + n3 = (1 + 2 + 3 + 4 + … + n)2.
7. Multiplication Table.
8. Division is the method of finding how often one given number,
called the Divisor is contained in another given number, called
the Dividend. The number expressing the times the divisor is
contained in the dividend is called the Quotient.
9. Long Division: When the divisor is greater than 20, the process is
called Long Division.
Divisor Dividend Quotient
536 870,42 162
536
3344
3216
1282
1072
210 Remainder
The least number consisting of figures from the left of the
dividend in which the divisor 536 is contained in 870 is called the
first partial dividend. The next figure in the dividend 3344 is called
the second partial dividend and 1282 is the third partial dividend. The
last figure, which should be less than the divisor, is called the
Remainder. This is clear that 87042 = 536 ´ 162 + 210.
10. Dividend = Divisor ´ Quotient + Remainder.
11. Division by factors (successive division); complete remainder.
Let us divide 57613 by 210, using factors and explain the rule to
find the remainder.
Now
210 = 5 ´ 6 ´ 7.
If we take 5, 6, 7 as d1, d2, d3 and 3, 2,
2, as r1, r2, r3.
Remainder = 3 + 5 ´ 2 + 5 ´ 6 ´ 2
= 3 + 10 + 60
\
Quotient = 274
57613 = 210 ´ 274 + 73.
5 5, 7, 6, 1, 2,
6 1, 1, 5, 2, 2, …3
7 1, 9, 2, 0,
…2
2, 7, 4,
…2
Numbers and Simplifications
3
The remainder obtained by the Division by factors method is
called the complete remainder or true remainder.
Þ Complete remainder
= r1 + d1r2 + d1 d2 r3.
12. Metric System: The Government of India has introduced the
metric system of weights and measures throughout the country.
This system derives its name from the word "Metre" which is the
standard unit of length in this system. The advantage of the
metric system is the great simplification of calculation in
different spheres of work.
In this system, the various units of length, area (surface) volume,
capacity and weight (mass) always bear a strictly decimal
relation to each other.
The international names of the five main units in the metric
system of weights and measures are:
Length measure unit is a Metre.
Area measure unit is a square metre.
Volume measure unit is a cubic metre.
weight measure unit is a Gram.
Capacity measure unit is a Litre.
An are contains 100 sq. metres and
A Hectare contains 100 areas or 10,000 sq. metres.
A cubic metre of volume contains 1000 litres or 1 kilo litre.
13. Vulgar Fractions: A fraction is represented by two numbers
written one above the other and sperated by a horizontal line.
Thus the fraction two-fifths is written as 2/5. The upper number
is called the numerator where as the lower number is called the
denominator. The numerator and the denominator of a fraction
are its terms.
A fraction is zero when its numerator is zero alone. The
denominator of a fraction is always non-zero. Fractions such as
8 7
,
…etc.
11 25
are called common or vulgar fractions. The value of the vulgar
fraction is not altered by multiplying or dividing by the
numerator and the denominator by the same number.
14. If the numerator and the denominator are large numbers, or if
their common factors cannot easily be guessed, we may find their
H.C.F.
3/5,
4
Subjective Arithmetic
15. A fraction is said to be a proper fraction if its numerator is less than
9 19
,
are all proper fractions. An
13 23
improper fraction is the one whose numerator is equal to N greater
its denominator. Thus 5/7,
7 7 15
, ,
are improper fractions. A
7 3 11
mixed fraction is one which consists of a whole number and a
than its denominator. Thus
fraction. Thus 2 71 , 15 23 are mixed fractions.
Complex Fraction is the one in which the numerator or
denominator or both are fractions.
3
3 5
+
4
15
4
7 are complex fractions.
Thus
,
,
7/9 2/3 2 1
–
5 4
16. Continued Fractions:
1
4+
3+
1
1
3+
4
1
or
3-
2
7+
4
5¸
3
4
are called continued fractions. To simplify such fractions, begin
at the bottom and work upwards.
17. Decimal Fractions: A decimal fraction or a decimal is a fraction
which has 10 or any power of 10 for its denominator and is
expressed in the decimal system of notation. A decimal fraction
is not altered by annexing ciphers to the right of the last figure.
Thus 0.567 and 0.56700 are equal.
18. Addition of decimals: Write down the number under one
another, placing units under units etc. then add as in the case of
integers, and place the decimal point under the points in the
given numbers. e.g. Add together 5.406, 0.8, 10.003 and 50.
5.406
0.8
10.003
50
66.209 Ans.
Numbers and Simplifications
5
19. SIMPLIFICATION: of vulgar fractions. (BODMAS).
In questions on fractions, signs ¸, ´, +, –,’ of (means
multiplication) and brackets are often involved. In simplifying
these questions the following order B O D M A S i.e.
(i) Remove the brackets (if any).
(ii) Then numbers or fractions which are connected by ‘of’ should
be simplified.
(iii) Then the division and multiplication in order must be carried
out and
(iv) Lastly the operations of addition (plus) or subtraction
(minus) should be performed.
20. SIMPLIFICATION: by Algebric formulae.
Remember the following.
1.
x2 - y2
x2 - y 2
= x + y and
= (x – y).
x-y
x+y
2. (x + y)2 – (x – y)2 = 4xy
3.
4.
5.
(x + y)2 + ( x - y)2
(x 2 + y 2 )
x3 + y 3
x 2 - xy + y 2
x3 - y3
x 2 - xy + y 2
= 2.
= (x + y)
= (x – y)
6. (x + y)3 = x3 + y3 + 3xy (x + y)
7. (x – y)3 = x3 – y3 – 3xy (x – y).
8. x2 + y2 + z2 – xy – yz – zx =
9.
1
[(x – y)2 + (y – z)2 + (z – x)2]
2
x 3 + y 3 + z 3 - 3 xyz
= (x + y + z)
( x 2 + y 2 + z 2 zx – yx - yz)
10. If x + y + z = 0, then
(i) x3 + y3 + z3 = 3xyz
(ii)
x3 + y 3 + z3
= 1.
3 xyz
6
Subjective Arithmetic
SOLVED EXAMPLES
Example 1. In a division sum, the quotient is 195, the divisor is equal
to the sum of the quotient and the remainder. Find the dividend.
Solution: We know that
Dividend = Divisior ´ Quotient + Remainder
Remainder = 195
Quotient = 105
Divisor = 195 + 105 = 300
\
Dividend = 300 ´ 105 + 195
= 31500 + 195
= 31695 Ans.
Example 2. A number when divided by 899 gives a remainder 63. What
will be the remainder, when the same number be divided by 29?
Solution: A number when divided by 899 gives a quotient say ‘Q’.
The remainder is given to be 63
Þ Such a number = 899 Q + 63
…(i)
Expressing this as multiple of 29, we have
The number = 29 ´ (31Q) + (2 ´ 29) + 5
\ The remainder obtained by dividing this number by 29 is 5.
Ans. 5.
Example 3. A certain number x is divided by 385 by dividing by three
prime factors. The quotient is 102, the first remainder is 4, the second
remainder is 6 and the third is 10.
Solution:
385 = 5 ´ 7 ´ 11.
Let x be the required number. Y and Z be the respective quotients.
X = 5Y + 4
…(i)
5 X
Y = 7Z + 6
…(ii)
7 Y
—4
Z = 11 ´ 102 + 10
…(iii) 11 Z
—6
= 1122 + 10 = 1132.
Y = 7z + 6
= 7 ´ (1132) + 6
= 7924 + 6 = 7930
102 — 10
Subjective Arithmetic : For All
Competitive Exams
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Publisher : Cosmos Bookhive
ISBN : 9788177290394
Author : S L Gulati
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