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*Basic Rules of Simplification*

*The Quantitative Aptitude section of the Bank exam consists of questions such as Simplification, Number Series, Permutation & Combination, Quadratic Equation, Data Interpretation, Data Analysis and other Miscellaneous questions.*

*Here is a short study-guide to help you crack questions on “Simplification and Approximation“*

*BODMAS Rule*

*It defines the correct sequence in which operations are to be performed in a given mathematical expression to find the correct value. This means that to simplify an expression, the following order must be followed –*

*B = Bracket,*

*O = Order (Powers, Square Roots, etc.)*

*D = Division*

*M = Multiplication*

*A = Addition*

*S = Subtraction*

*⇒To solve simplification questions correctly, you must apply the operations of brackets first. Further, in solving for brackets, the order – (), {} and [] – should be stricly followed. *

*⇒Next you should evaluate exponents (for instance powers, roots etc.)*

*⇒Next, you should perform division and multiplication, working from left to right. (division and multiplication rank equally and are done left to right).*

*⇒Finally, you should perform addition and subtraction, working from left to right. (addition and subtraction rank equally and are done left to right).*

**EXAMPLE : 12 + 22 ÷ 11 × (18 ÷ 3)^2 – 10***= 12 + 22 ÷ 11 × 6^2 – 10 (Brackets first)*

*= 12 + 22 ÷ 11 × 36 – 10 (Exponents)*

*= 12 + 2 × 36 – 10 = 12 + 72 – 10 (Division and multiplication, left to right)*

*= 84 – 10 = 74 (Addition and Subtraction, left to right)*

**BRACKETS***The various kind of brackets are:*

*(i) ‘–’ is known as line (or bar) bracket or vinculum.*

*(ii) ( ) is known as parenthesis, common bracket or small bracket.*

*(iii) { } is known as curly bracket, brace or middle bracket.*

*(iv) [ ] is known as rectangular bracket or big bracket.*

*The order of eliminating brackets is:*

*(i) line bracket*

*(ii) small bracket (i.e., common bracket)*

*(iii) middle bracket (i.e., curly bracket)*

*(iv) big bracket (i.e., rectangular bracket)*

*Note:*

*(i) Even + Even = Even*

*Even – Even = Even*

*Even × Even = Even*

*Even ÷ Even = Even*

*(ii) Odd + Odd = Even*

*Odd – Odd = Even*

*Odd × Odd = Odd*

*Odd ÷ Odd = Odd*

*(iii) Even + Odd = Odd*

*Even – Odd = Odd*

*Even × Odd = Even*

*Even ÷ Odd = Even*

** NUMBER SYSTEM **

*Types of Numbers:** 1. Natural Numbers: A number n > 0 where n is counting number; [1,2,3…]** 2. Whole Numbers: A number n ≥ 0 where n is counting number; *0,1,2,3…+.** 0 is the only whole number which is not a natural number.** Every natural number is a whole number.** 3. Integers: A number n ≥ 0 or n ≤ 0 where n is counting number;…,-3,-2,-1,0,1,2,3… are integers.** 4. Positive Integers: A number n > 0; [1,2,3…]** 5. Negative Integers: A number n < 0; [-1,-2,-3…]** 6. Non-Positive Integers: n ≤ 0; *0,-1,-2,-3…]** 7. Non-Negative Integers: A number n ≥ 0; *0,1,2,3…+** 0 is neither positive nor negative integer.** 8. Even Numbers: A number divisible by 2; [for example 0,2,4,…]** 9. Odd Numbers: A number not divisible by 2; [for example 1,3,5,…]** 10. Prime Numbers: A number numbers which is divisible by themselves only apart from 1.** 1 is not a prime number.*

** DIVISIBILITY**

*1. Divisibility by 2:A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.** Example: 64578 is divisible by 2 or not?** Step 1 – Unit digit is 8.** Result: 64578 is divisible by 2.*

*2. Divisibility by 3: A number is divisible by 3 if sum of its digits is completely divisible by 3.** Example: 64578 is divisible by 3 or not?** Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30** which is divisible by 3.** Result: 64578 is divisible by 3.*

*3. Divisibility by 4: A number is divisible by 4 if number formed using its last two digits is completely** divisible by 4.** Example: 64578 is divisible by 4 or not?** Step 1 – number formed using its last two digits is 78** which is not divisible by 4.** Result: 64578 is not divisible by 4.*

*4. Divisibility by 5: A number is divisible by 5 if its unit digit is 0 or 5.** Example: 64578 is divisible by 5 or not?** Step 1 – Unit digit is 8.** Result: 64578 is not divisible by 5.*

*5. Divisibility by 6: A number is divisible by 6 if the number is divisible by both 2 and 3.** Example: 64578 is divisible by 6 or not?** Step 1 – Unit digit is 8. Number is divisible by 2.** Step 2 – Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30** which is divisible by 3.** Result: 64578 is divisible by 6.*

*6. Divisibility by 7:A number of 2 digits is divisible by 7 because 3 × 6 + 3 = 21. 21 is divisible by 7.** A number of three or more digits is divisible by 7 if the sum of the numbers formed by the last two digits and twice the number formed by the remaining digits is divisible by 7.** For Example:** (i) 574 is divisible by 7 because 74 + 5 × 2 = 74 + 10 = 84 is divisible by 7.** (ii) 2268 is divisible by 7 because 68 + 22 × 2 = 68 + 44 = 112 is divisible by 7.*

*7. Divisibility by 8: A number is divisible by 8 if number formed using its last three digits is completely** divisible by 8.** Example: 64578 is divisible by 8 or not?** Step 1 – number formed using its last three digits is 578** which is not divisible by 8.** Result: 64578 is not divisible by 8.*

*8. Divisibility by 9: A number is divisible by 9 if sum of its digits is completely divisible by 9.** Example: 64579 is divisible by 9 or not?** Step 1 – Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31** which is not divisible by 9.** Result: 64579 is not divisible by 9.*

** SQUARE ROOT AND CUBE ROOT**

**Main Concepts and Results**

*• A natural number is called a perfect square if it is the square of some natural number. i.e., if m = n2, then m is a perfect square where m and n are natural numbers.*

*• A natural number is called a perfect cube if it is the cube of some natural number. i.e., if m = n3, then m is a perfect cube where m and n are natural numbers.*

*• Number obtained when a number is multiplied by itself is called the square of the number.*

*• Number obtained when a number is multiplied by itself three times are called cube number.*

*• Squares and cubes of even numbers are even.*

*• Squares and cubes of odd numbers are odd.*

*• A perfect square can always be expressed as the product of pairs of prime factors.*

*• A perfect cube can always be expressed as the product of triplets of prime factors.*

** ****Square Root**

**Example: ****Square root of 3481**

*Step 1 :- Split number in two parts i.e. 34 and 81(last two digits)*

*Step 2 :-We know that square of number ends in 1, so square root ends either in 1 or 9.*

*Step 3:-Check, 34 lies between 25 (square of 5) and 36 (square of 36). Take smaller number.*

*So, our answer is either 51 or 59.*

*but we know 502 = 2500 and 602 = 3600, 3481 is nearest to 3600. So the answer is 59.*

**Ex4: 1000***Step 1 :- 961(312) < 1000 < 1024(322)*

*Now, 1000 is nearest to 1024*

*So, 32 – ((1024-1000)/(2× 32))*

*32 – (24/64)*

*32-.375 = 31.625*

*or 31+((1000-961)/(2× 31))*

*31 + (39/62)*

*31+.629 ≈ 31.63*

** Cube root**

**Example: cube root of 3112136***Split in two parts 3112 136*

*Number will end with 6*

*143 (2744) < 3112 < 153 (3375)*

*Choose the smaller number and answer will be 146.*

**Square Root Upto 25:-**

**Percentage Type**

*Common Percentage method:-*

*12 % of 555 + 15 % of 666= ?*

*Here 10 % is common 10 % of (555+666)=10 % of 1221=122.10** First part (10+2)=2% of 555= 11.10** Second part (10+5)=5% of 666= 33.30** So (122.10+11.10+33.30)= 166.50*

*Breaking Method:-*

*68 % of 4096 +72% of 5120 -23 % of 6931 -17 % of 1341= ?(Approximate)*

*(60+8)%of 4096 +(70+2) % of 5120 –(20+3)% of 6931 –(10+7)% of 1341** =2456.00 +327.68 +3584.00+102.40 -1386.20-207.93-134.10-93.87** =6470.08 -1821.47** =4648.61** ≈450*

*Unit digit Method:-*

*75 % of 4860 = ?** =(300/4)% of 4860** =3*1215** =3645*

*Base method:-*

*95% of 4860 = ?*

*=(100-5)% of 4860** =4860-243** =4617*

*Whole Number method:-*

*139.001 % of 1299.99 + 159.99 % of 1359.99= ? (Approximate)*

*(140 *1300)/100 + (160 *1360)/100** =1820+2176=3996*

*Fraction method:-*

*125 of 488 + .625 of 824 = ?*

*=12.5 % of 488 + 62.5 % of 824** =1/8 *488 + 5/8 * 824** =61 + 515** =576*

** ADDITION AND SUBTRACTION**

*I. Addition of smaller number to larger number is easier than addition of larger number to smaller number*

*Example :- addition in the order 5817 + 809 + 67 + 8 is easier than the addition in the order 8 + 67 + 809 + 5817. Hence to add the numbers, it is better to first arrange them in decreasing order and then add them.*

*II. To find the sum like 6345 + 2476 + 802, first add the thousands and then hundreds, tens and once in order.*

*Thus** 6345 + 2476 + 802*

*= 6000 + 2000 ( = 8000) + 300 (= 8300)+ 400 (= 8700) + 800 (= 9500)+ 40 (= 9540) + 70 (= 9610) + 5 (= 9615) + 6 (9621) + 2*

*= 9623*

** Addition Shortcut**

*Example:- 116 + 39*

*(Here we can write this 39 as 40-1)** = 116 + (40 – 1)** = 116 + 40 – 1** = 156 – 1 (Instead of adding 39 to 116, we just add 40 to 116 (because we can do this without using pen and paper) and later we subtract one from it)** = 155*

*Example:- 116 + 97** = 116 + (100 – 3)** = 116 + 100 – 3 (Here, instead of adding 97 to 116, we are just adding a 100 to 116 and then subtracting 3 from it .** = 216 – 3** = 213*

** H.C.F and L.C.M**

*H.C.F is the highest common factor or also known as greatest common divisor, the greatest number** which exactly divides all the given numbers.*

*Follow the steps below to find H.C.F of given numbers by prime factorization method.*

*1. Express the given numbers as product of their prime factors** 2. Check for the common prime factors and find least index of each common prime factor in the given numbers** 3. The product of all common prime factors with the respective least indices is H.C.F of given numbers.*

*Example-** H.C.F of 12, 36, 48*

*Solution-*

*1. Express the numbers as product of prime factors*

*12= 3 * 2^2*

*36=3^2 * 2^2*

*48=3 * 2^4*

*2. The common prime factors are 2 and 3 and the corresponding least indices are 2 and 1 respectively** 3. The product of all the common prime factors with the respective least indices*

*H.C.F of 12, 36, 48 =2^2 *3= 12*

*Least Common Multiple (L.C.M)** L.C.M is least common multiple, the smallest number which is exactly divisible by all the given numbers.*

*Follow the steps below to find L.C.M of given numbers .** 1. Express the given numbers as product of their prime factors.** 2. Find highest index in all the prime factors of given numbers.** 3. The product of all the prime factors with respective highest indices is the L.C.M of given numbers.*

*Example:** L.C.M of 14, 42, 36*

*Solution:-*

*1.Express the numbers as product of prime factors.*

*14= 2*7** 36= 3^2 *2^2** 42 = 2*3*7*

*2. The highest index of 2, 3, 7 are 2, 2, 1 respectively** 3. The product of all the prime factors with the respective highest indices.*

*L.C.M of 14, 36, 42 =2^ 2 *3^2 *7 = 252*

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