**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – Number System**

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In Hindu Arabic System, we use ten symbols 0,1,2, 3,4, 5,6, 7,8,9, which are called digits to represent any number. Out of these ten digits, 0 is called an insignificant digit whereas the others are called significant digit.

A group of digits representing a number is called a numeral. To write a number, we put digits from right to left at the places designated as unit, ten, hundred, thousand, ten thousand, lakh, ten lakh and so on.

Now, let us see how the number 5 1 287254 is represented.

It is read as “Five crore twelve lakh eighty seven thousand two hundred fifty four”.

**Value of Numerals**

**1. Face Value** Face value of a digit is the value of the digit itself irrespective of its place in the numeral.

**e.g.,** In number 41232 => Face value of 3 is 3.

**2. Place Value** The place value of a digit is the place at which digit is present in the numeral

**e.g., I**n number 41232 => Place value of 3 is 3 x 100 = 300.

**Classification of Numbers**

**Natural Numbers** All counting numbers, except zero (0) are called natural numbers denoted by N. e.g., 1, 2, 3,… etc.

**Whole Numbers** All natural numbers along with zero are called whole numbers denoted by W. e.g., 0, 1, 2, 3,… etc.

**Even Numbers** All counting numbers which are divisible by 2 are called even numbers, e.g., 2, 4, 6, 8,… etc.

**Odd Numbers** All counting numbers which are not divisible by 2 are called odd numbers, e.g., 1, 3, 5,… etc.

**Prime Numbers** A counting number is called prime number, if it has only two factors i.e., 1 and the number itself, e.g., 2, 3, 5, 7, 11, 13,… etc,

**Co-prime Numbers** Two natural numbers are said to be co-primes, if their highest common factor is 1. e.g., (9,2), (5,6), (11, 15), …etc. Co-prime numbers are also called as relatively prime numbers.

**Composite Numbers** A composite number is one which has other factors besides 1 and itself. Thus, it is a non-prime number, e.g., 4, 6, 9, 14, 15 etc.

**Note** 1 is neither prime nor composite.

**Integers** The collection of all whole numbers along with the negative numbers are called integers. The set of integers is denoted by Z or I. e.g., -4, -2, 0,-1, 3, 4,….

Here, -4, -2 are called negative integers and 1, 3,4 are called positive integers.

**Example** If p and q are two relatively prime positive integers such that p + q= 10, p < q, then the number of possible values of p is **SSC (10+2) 2012**

(a) 1

(b) 2

(c) 3

(d) 4

**Answer:**

(a) Two numbers are said to be co-prime, if their HCF is 1.

Co-prime numbers can be prime or not,

Now, p + g = 10, p< q

Possible values of p are, p = 3, q = 7

Possible values of p is 1 values.

**Rational Numbers** A number which can be expressed in the form of p/q, where p and q are integer and q ≠ 0 is q called rational number.

**Irrational Numbers** A number which cannot be expressed in the form of p/q, where p and q are integer and q ≠ 0. e.g., √2, √3, √7,… etc.

**Real Numbers** Real numbers comprises of both rational and irrational numbers, e.g., 7/9, √2, √5, π, 8/9,… etc.

**Divisibility Test**

A number is said to be

**Divisible by 2** When its unit’s place digit is even or zero. Divisible by 3 When the sum of its digits is divisible by 3.

**Divisible by 4** When the number formed by the last two digits is divisible by 4.

**Divisible by 5** When its unit’s place digit is either 0 or 5.

**Divisible by 6** When it is divisible by 2 and 3 both.

**Divisible by 7** When difference between twice the digits at ones place and the number formed by other digits is either zero or multiple of 7.

e.g., 581 is divisible by 7 because 58-2×1 = 56

As, 56 is divisible by 7.

581 is divisible by 7.

**Divisible by 8** When number formed by last three digits is divisible by 8.

**Divisible by 9** When sum its digits is divisible by 9. Divisible by 10 When unit’s place digit is zero.

**Divisible by 11** When the sum of digits at odd places and even places are equal or differ by a number divisible by 11.

**Divisible by 12** When it is divisible by 3 and 4 both.

**Important Divisibility Rules**

• If a number is divisible by another number, then it is divisible by each of the factors of that number.

• If a number is divisible by two co-prime numbers, then it is divisible by their product also.

• If two given numbers are divisible by a number, then their sum and difference is also divisible by that number.

• (x^{n} – a^{n}) is divisible by (x – a) for all values of n.

• (x^{n} – a^{n}) is divisible by (x + a) for even values of n.

• (x^{n} + a^{n}) is divisible by (x + a) for odd values of n.

**Division of Numbers**

**(Division Algorithm Lemma)**

Let a and b be two integers such that b ≠ 0. On dividing a by b, q will be quotient and r will be the remainder, then the relationship between a, b, q and r is established as

a = bq + r

In general, we have .

Dividend = Divisor x Quotient + Remainder

**Example** 47 is added to the product of 71 and an unknown number. The new number is divisible by 7, giving the quotient 98. The unknown number is a multiple of **ssc (10+2) 2011**

(a) 2

(b) 5

(c) 7

(d) 3

**Example** A number, when divided by 114, leaves remainder 21. If the same number is divided by 19, then the remainder will be **ssc (10+2)2010**

(a) 1

(b)2

(c) 7

(d) 17

**To Find the Unit’s Place Digit in the Product of Numbers**

To find the unit’s place digit in the given product, first take unit’s place digits of each number. Find the product of these digits. If there is any ten place digit, then take unit’s place digit to continue the product. Then, the unit’s place digit in the last number is the required digit.

**Simplification**

The word simplification refers to a procedure useful for converting a lengthy expression into simplest form.

Rule for solving question on simplification is VBODMAS. You have to solve the expression by solving various operations in a particular sequence given by VBODMAS. It means

V-Underline portion i.e., bar

B-Brackets should be removed in the order (), {} and [ ], i. e., operations inside the bracket should be done first.

O-Of [whereever ‘of is given, the meaning is X or multiplication]

D- Division

M- Multiplication

A- Addition

S- Subtraction

**Fraction**

A number expressed in the form of a/b where a and b are integers and b ≠ 0, where a is known as numerator and b is known as denominator.

**Types of Fraction**

**Proper Fraction** When numerator is less than denominator, then fraction is called proper fraction.

**Improper Fraction** When numerator is greater than denominator, then fraction is called improper fraction.

**Like Fraction** A number of fractions having same denominator are called like fraction.

**Unlike Fraction** A number of fraction having different denominators are called unlike fraction.

**Compound Fraction** It is fraction of fraction.

e.g, ,1/4 of 5/7 of 7/9 etc

**Complex Fraction** In such fractions numerator or denominator both are fractions.

eg,

**Mixed Fraction** Those fractions which consists a whole number and a proper fraction are know as mixed fraction.

eg.,3 1/4,2 1/2 etc

**Continued Fraction** It contains an additional fraction in the numerator or in the denominator.

eg.,

**Example** The denominator of a fraction is 3 more than its numerator. If the numerator is increased by 7 and the denominator is decreased by 2, we obtain 2. The sum of numerator and denominator of the fraction is **SSC (10 + 2) 2011**

(a) 5

(b) 13

(c) 17

(d) 19

**Example** A number whose one-fifth part increased by 4 is equal to its one-fourth part diminished by 10, is **SSC (10 + 2) 2011**

(a) 260

(b) 280

(c) 240

(d) 270

**Decimal Fraction** In this type of fraction, denominator has power of 10. There is a special type of decimal fraction called ‘Recurring decimal fraction’.

**Recurring Decimal Fraction** This is the decimal fractions in which one or more decimal digits are repeated again and again.

e.g,

There are two types of recurring decimal fraction

**1. Pure Recurring Decimal Fraction** When all the figures in a decimal fractions are repeated after the decimal point, then the decimal fraction is called a pure recurring decimal fraction.

**2. Mixed Recurring Decimal Fraction** When some figures are not repeated while some of them are repeated in recurring decimal, then it is called mixed recurring decimal fraction.

**Essential Points**

- Sum of first n natural numbers = n(n + 1)/2
- Sum of first n odd numbers = n
^{2} - Sum of first n even numbers = n(n + 1)
- Sum of square of first n natural numbers =n(n+1)(2n+1)/6
- Sum of cube of first n natural numbers =[n(n+1)/2]
^{2} - (x
^{m}– a^{m}) is divisible by (x – a) for all values of m. - (x
^{m}– a^{m}) is divisible by (x + a) for even values of m. - (x
^{m}+ a^{m}) is divisible by (x + a) for odd values of m. - Product of two numbers=HCF of numbers x LCM of the numbers.

**Reference Corner**

**1. What is the value of 3/4 + 8/9?**

(a) 57/27

(b) 11/13

(c) 59/36

(d) 11/9

**Answer:**

**2. The difference between the greatest and the least four digit numbers that begins with 3 and ends with 5 is SSC (10 + 2) 2015**

(a) 900

(b) 909

(c) 999

(d) 990

**Answer:**

**3. The least value of n, such that (1 + 3 + 3 ^{2} + … + 3^{n}) exceeds 2000, is SSC (10 + 2) 2014**

(a) 7

(b) 8

(c) 5

(d) 6

**Answer:**

**4. The simplified value of (0.2) ^{3} x 200 ÷ 2000 of (0.2)^{2} is SSC (10 + 2) 2014**

(a) 1/10

(b) 1

(c) 1/100

(d) 1/50

**Answer:**

**5. The odd one out from the sequence of numbers 19, 23, 29, 37, 43, 46, 47 is SSC (10 + 2) 2014**

(a) 37

(b) 19

(c) 23

(d) 46

**Answer:**

**6. The next number of the sequence 1/2, 3/4, 5/8, 7/16, … is SSC (10 + 2) 2014**

(a)9/24

(b)9/32

(c)10/24

(d)11/32

**Answer:**

**7.The least number by which 20184 must be multiplied, so as to make the product a perfect square is SSC (10 + 2) 2014**

(a)5

(b)6

(c)2

(d)3

**Answer:**

**8. A teacher wants to arrange his students in an equal number of rows and columns. If there are 1369 students, the number of students in the last row are SSC (10 + 2) 2014**

(a) 37

(b) 33

(c) 63

(d) 47

**Answer:**

**9. The first term of an arithmetic progression is 22 and the last term is -11. If the sum is 66, the number of terms in the sequence are SSC (10+2) 2014**

(a) 10

(b) 12

(c) 9

(d) 8

**Answer:**

**10. In a division sum, the divisor is 3 times the quotient and 6 times the remainder. If the remainder is 2, then the dividend is SSC (10+2) 2013**

(a) 36

(b) 28

(c) 50

(d) 48

**Answer:**

**11. 1/7 + (999 692/693) is equal to SSC (10+2) 2013**

(a) 99800

(b) 99900

(c) 1

(d) 99000

**Answer:**

**12. The number of prime factors in 6 ^{333} x 7^{222} x 8^{111} is SSC (10 + 2) 2013**

(a) 1111

(b) 1211

(c) 1221

(d) 1222

**Answer:**

**Practice Exercise**

**1.Amit, while solving his Maths homework, took 12 as divisor instead of 21, in a question on division with zero remainder. The quotient obtained by him was 35. Find the correct quotient.**

(a) 10

(b) 12

(c) 20

(d) 15

**2.The sum of three consecutive odd numbers is 20 more than the first number among three. What is the middle number?**

(a) 7

(b) 8

(c) 12

(d) 9

**3.The sum of series 1 + 2+ 3+ 4+ …+ 998 + 999 + 1000**

(a) 5050

(b) 500500

(c) 550000

(d) 55000

**4.On dividing a certain number by 357, the remainder is 39. On dividing the same number by 17. What will be the remainder?**

(a) 5

(b) 3

(c) 7

(d) 6

**5.Each member of a kitty party contributed twice as many rupees as the total collection was Rs 3042. Find the number of members present in the party.**

(a) 22

(b) 32

(c) 42

(d) 39

**6. The difference between a number of two digit and the number obtained interchanging its digits is 63. What is the difference between its digits?**

(a) 5

(b) 6

(c) 7

(d) 8

**7. Sumit, Aditya and Sandeep cycling round a circle 1 km in circumference at they rate of 10, 20 and 40 m/min, respectively. If they all start together in the same direction, when will they again be together at the same place?**

(a) After 50 min

(b) After 240 min

(c) After 800 min

(d) After 100 min

**8. If the sum of the digits of a two digit number is 9 and the difference of those digits is 3, what is the product of the digits of the same number?**

(a) 9

(b) 36

(c) 18

(d) 72

**9. Out of fractions 5/7, 7/13, 4/7, 4/15 and 9/14 which is the second highest?**

(a) 5/7

(b) 7/13

(c) 4/7

(d) 4/15

**10. A number divided by 899 gives a remainder of 63. If the number is divided by 29, the remainder will be **

(a) 2

(b) 5

(c) 13

(d) 28

**11. The greatest number of five digit which when divided by 3, 5, 8, 12 have 2 as remainder**

(a) 99999

(b) 99958

(c) 99960

(d) 99962

**12. The sum of all natural numbers between 100 and 200 which are multiples of 3 is**

(a) 5000

(b) 4950

(c) 4980

(d) 4900

**13. Amar spends 1/3 of his income on food, 2/5 of his income on house rent and 1/5 of his income on clothes. If he still have Rs 400 left with him. Find Amar’s income, **

(a) Rs 4000

(b) Rs 5000

(c) Rs 6000

(d) Rs 7000

**14. A milkman has 75 L milk in one cane and 45 L in another. The maximum capacity of container which can measure mild of either container exact number of times, is**

(a) 1 L

(b) 5 L

(c) 15 L

(d)25 L

**Answers**

**Hints & Solutions**

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