UPKAR PRAKASHAN, AGRA-2

ByDr. Lal & Jain

(Useful for All Competitive Examinations)

Revised Edition

© Publishers

Publishers

UPKAR PRAKASHAN(An ISO 9001 : 2000 Company)

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● This book or any part thereof may not be reproduced in any form byPhotographic, Mechanical, or any other method, for any use, withoutwritten permission from the Publishers.

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ISBN 13 : 978-81-7482-058-7

Price : 250·00(Rs. Two Hundred Fifty Only)

Code No. 447

Printed at : UPKAR PRAKASHAN (Printing Unit) Bye-pass, AGRA

Contents

1. Numbers ………………………………………………… 3—252. H.C.F. and L.C.M. ……………………………………... 26—423. Fraction………………………………………………… 43—614. Simplification………………………………………….. 62—795. Unitary Method………………………………………… 80—986. Percentage…………………………………………….. 99—1237. Work and Time………………………………………. 124—1448. Pipes and Cisterns……………………………………. 145—1579. Distance and Time & Questions on Trains………….. 158—182

10. Problems Based on Boats and Streams……………… 183—19511. Ratio, Proportion and Proportional Division………… 196—21912. Partnership…………………………………………… 220—23313. Square Root and Cube Root…………………………. 234—25114. Average……………………………………………… 252—27515. Profit and Loss………………………………………. 276—31016. Simple Interest………………………………………. 311—33717. Compound Interest………………………………….. 338—36618. Alligation…………………………………………….. 367—38419. Area………………………………………………….. 385—41420. Cuboid and Cube…………………………………….. 415—43021. Cylinder and Cone…………………………………… 431—44822. Sphere……………………………………………….. 449—46823. Stocks and Shares…………………………………… 469—48524. Data Sufficiency……………………………………… 486—50225. Interpretation of Data………………………………… 503—56426. Statistics……………………………………………… 565—58727. To Find the Wrong Term in A Number of Series…… 588—60428. Problems Based on Ages…………………………….. 605—624

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1 NumbersIntroductory01. The numbers written by the symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are

known as digits.02. The numbers which are divisible by 2 such as

2, 4, 6 etc. are known as even numbers.03. The numbers which are not divisible by 2 such as

1, 3, 5, etc. are known as odd numbers.04. The numbers 1, 2, 3,…… which are used for counting are known as the

natural numbers.Thus 528 is a natural numbers but 0 – 3, 14.6, and 3/4 etc. are notnatural numbers.

05. If zero is included in natural numbers the set so obtained is known aswhole numbers. For example—

0, 1, 2, 3,… etc. are whole numbers but –1, –2, 3, 2 19 are not whole

numbers.06. The whole numbers positive as well as negative are known as integers.

For example—

–7, –6, –5,…0, 1, 2, 3, 4 etc. are integers but √⎯ 5, 0.32, – 1.2 and 13 etc.

are not integers.07. The numbers which can be expressed in the form of p/q, where p and q

are integers but q ≠ 0, are known as rational numbers. For example—

25, – 62, 37265

, – 23, 0 etc. are rational numbers.

08. A number which cannot be expressed in terminating decimal is knownas an irrational number. For example—

√⎯ 5, 3 + √⎯ 2, etc. are irrational numbers.09. The numbers which are either rational or irrational are known as real

numbers. For example—

3, –2, √⎯ 7 etc. are real numbers.10. A natural number other than 1, is a prime number if and only if it is

divisible by 1 and the number itself. For example—2 is a prime number but 6 is not a prime number.

4A | O. Math

11. The natural number other than 1 is a composite number which isdivisible by 1, the number itself and atleast one more other number. Forexample 6 is a composite number because it is divisible by 1, 2, 3 and 6.

Note— 1. is neither prime number nor composite,

2. is only the number which is prime as well as even number

12. Dividend = (Divisor × Quotient) + Remainder.

13. In division performed by factors

True remainder = First remainder + (First divisor × Second remainder) +(First divisor × Second divisor × Third remainder) +……For example on dividing 305 by 15, what will be the remainder.

3 307

5 102 – 1

20 – 2

∴ True remainder = 1 + (3 × 2)= 1 + 6= 7

14. The face value of a digit is known as the intrinsic value. For examplethe intrinsic value of 8 in 9872 is 8.

15. The value of a digit according to its particular place is known as placevalue. For example the place value of 8 in 9872 is 800.

Test of Divisibility01. A number is divisible by 2 if it ends in zero or in an even digit which is

a multiple of 2. For example

30, 36 etc. are divisible by 2

02. A number is divisible by 3 if the sum of its digits is divisible by 3. Forexample sum of the digits of 402 is 6 and 6 is divisible by 3. Therefore,402 is also divisible by 3.

03. A number is divisible by 4 if the last two digits i.e. tens and units aredivisible by 4. For example last two digits of 372 are 72 and 72 isdivisible by 4. Therefore, 372 is also divisible by 4.

04. A number is divisible by 5 if it ends in zero or 5. For example

205, 320 etc. are divisible by 5.

05. A number is divisible by 6 if it is divisible by 2 as well as by 3. Forexample 510 is divisible by 2 as well as by 3. Therefore, 510 is divisibleby 6.

06. To check whether the given number is divisible by 7, the unit digit ofthe given number is doubled and then it is subtracted from the numberobtained after omitting the unit digit. If the remainder is divisible by 7,then the given number is also divisible by 7.

O. Math | 5A

If the given number is large enough, then the process is repeated tillsuch a small number is obtained which can be easily checked whether itis divisible by 7.

Example 1.343 is to be checked whether it is divisible by 7.

Solution :On doubling the unit digit 3 of 343, we get 6 then 34 – 6 = 28Since 28 is divisible by 7, hence, 343 is divisible by 7.

Example 2.5499838 is to be checked whether it is divisible by 7. The unit digit of

the given number is 8.

Solution :

On doubling 8 and subtracting from the number which is obtained afteromitting the unit digit, we get

549983 – 16 = 549967

54996 – 2 × 7 = 54982

5498 – 2 × 2 = 5494

549 – 2 × 4 = 541

54 – 2 × 1 = 52

Since 52 is not divisible by 7, therefore, 5499838 is also not divisibleby 7.

07. A number is divisible by 8 if its last three digits i.e., hundreds, tens andunits, are divisible by 8. For example the last three digits of 5248 are248 and 248 is divisible by 8, hence 5248 is also divisible by 8.

08. A number is divisible by 9 if the sum of its digits is divisible by 9. Forexample the sum of the digits of 576 is 18 and 18 is divisible by 9.Therefore, 576 is also divisible by 9.

09. A number is divisible by 10 if it ends in zero. For example the last digitof 410 is zero, therefore, 410 will be divisible by 10.

10. A number is divisible by 11 if the difference between the sums of thedigits in the even and odd places is zero or a multiple of 11. Forexample in 1573, the sum of the digits in odd places is 5 + 3 i.e., 8 andthe sum of the digits in even places is 7 + 1 i.e 8. Therefore, thedifference between these sums is zero. Hence 1573 is divisible by 11.

Similarly in 9174, the sum of the digits in odd places is 4 + 1 i.e. 5 andthe sum of digits in even places is 7 + 9 i.e. 16. Now the differencebetween these sums is 16 – 5 i.e. 11 which is a multiple of 11.Therefore, 9174 is divisible by 11.

Quicker Objective Arithmetic in English

Publisher : Upkar Prakashan ISBN : 9788174820587 Author : Dr. Lal & Jain

Type the URL : http://www.kopykitab.com/product/4223

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