SIMPLIFICATIONS

SIMPLIFICATIONS

All of you can answer these types of questions simply. But the fact is that you must answer these questions in the required time. For that you should have lot of practice along with studying.
TYPE -I
Basic Arithmetic Operations are Addition, Subtraction, Multiplication and Division.
I.ADDITION

1.
The addition of two positive numbers will always give a positive number.
Eg:- (a) 5 2 = 7 (b) 7 2 = 9
2.
The addition of two negative numbers will always give a negative number.
Eg:- (a) -2-4 = -6
(b) -7-2 = -9
3.
The result of addition of a positive number and a negative number will be the difference of these two numbers with the sign of the greater number.
Eg:- (a) -2 8 = 6
(b) -7 2 = -5
4.
If the sum of two numbers is 0, then both numbers are 'additive inverses' of each other.
Eg:- (-2) (2) = 0.
this (-2) is the additive inverse of (2) and vice versa.
In short
3 2 =5
-3-2 =-5
3-2 =1
-3 2 =-1
-3 3 =0
II.SUBTRACTION

To subtract a number from another number, the additive inverse of the number to be subtracted is added to the other number.
Eg:-7-(-3)= 73 = 10
III.MULTIPLICATION

1.The product of two positive numbers will always be a positive number.
Eg:- (1) 4 x 3 = 12
(2) 6 x 8 = 48
2.The product of two negative numbers will always be a positive number.
Eg:- (1) -5 x -2 =10
(2) -11 x -2 = 22
3.The product of a positive number and a negative number will always be a negative number.
Eg:- (1) 6 x -3 = -18
(2) -5x7 = -35
In short
3 x 2 =6
-3 x -2 =6
-3x2 =-6
-2x3 =-6
IV. DIVISION

1.
The division of two positive numbers will always be a positive number.
Eg:- (1) 36/ 6 = 6
(2)42 /3 = 14
2.
The division of two negative numbers will always be a positive number
Eg:- (1)-40/-8 = 5
(2)-30/ -10 = 3
3.
The division of a positive number and a negative number will always be a negative number.
Eg:- (1)-20 / 5 = -4
(2)60 /-10 = -6
In short
10 / 2= 5
(-10) /(-2)= 5
(-10) / 2= -5
10 (-2)= -5
V. VBODMAS

It is the rule of order of simplification
V indicates Vinculum or bar
B indicates bracket
O indicates order of
D indicates division
M indicates multiplication
A indicates addition and
S indicates subtraction
EXAMPLES

(3 6 x 7) 10 Of 6rd/3 of 30 / 6=?
Applying VBODMAS rule=>
=(342) x 10 of 6rd/3 of 5
= 450x 6/3 x 5 = 4500
2.If (x 3/2x10) = 60% of -1th/5 of (36/ 12 x 9), find the value of x.
(x 3/2 x 10)=60/100x1/5(3x9)
x 15=12/100*27
x=3.24-15
=-11.76
.'.x=-11.76
TYPE - II

*Sum of first 'n' natural numbers
=n(n l)/2
ie, 123 ..........n = n(n1)/2
*Sum of first 'n' odd number =n2
ie,135.......n=n2
*Sum of first 'n' even numbers = n(n 1)
ie, 2 4 6 ..........n = n(n 1)
*Sum of squares of first 'n' natural numbers = 1/6 n(nl)(2nl)
ie, 1^2 2^2 3^2 .... n2
=1/6 n(n 1)(2n 1)
*Sum of cubes of first 'n' natural numbers
=1/4n^2(n1)^2
ie, 1^3 2^3 3^3 .......n3 = 1/4n^2 (nl)2
EXAMPLES

1.
Find the sum of first 20 natural numbers.
=n(nl)/2
= 20(201) = 20x21/2 210
2.
Find the sum of first 15 odd numbers n2 = sum of the first 15 natural nos.
= 15^2 = 225
3.
Find the sum of first 10 even numbers
= n(nl)
= 10(101)
= 10 x 11 = 110
4.
What is the sum of squares of first 6 natural numbers?
1^2 2^2 3^2 4^2 5^2 6^2
= 1/6n(nl)(2n l)
= 1/6x6x7xl3=91
5.
What is the sum of cubes of first 5 natural numbers?
l^3 2^3 3^3 4^3 5^3 =1/4n^2(n1)^2
= 1/4x 25x36=225
TYPE -III

ALGEBRAI FORMULAE FOR SIMPLIFICATION
*(ab)2=a22abb2 (a-b)2=a2-2abb2 (ab)2=(a-b)2 4ab (a-b)2 =(ab)2-4ab
*(a b)(a-b) = a2-b2 ax(bc)=(a*b)(a*c) (abc)2 = a2b2c2 2ab2bc2ac
*(ab)3 = a33ab(ab)b3 (a-b)3=a3-3ab(a-b)-b3 a3b3=(ab)(a2-abb2) a3-b3 = (a-b)(a2 abb2)
*(a b)2(a-b)2=2(a2b2) (a b)2-(a-b)2 =4ab
EXAMPLES

1.
15x15 - 6x6/9x9-2x 9x 88x8 =
This is of the form a2-b2 /a2-2abb2
=(a b)(a-b)/(a-b)2
=(15 6)(l5-6)/(9-8)2= 21x9/1=189
2.
What is,
36752x36752-20796x20796/36752 20796
This is of the form a2-b2/ab
a2 -b2/ab= (a b)(a-b)/(ab)=(a-b)
= 36752 - 20796 = 15956
3.
Evaluate 3.7x4.5 6.3x4.5/2.3*8.9-2.3*7.9
This is of the form ax bx/cy-dy=(a b)x/(c-d)y
=(3.7 6.3)4.5 /(8.9-7.9)2.3 = 10x4.5/1x2.3
=450/23
4.
(6.65)^3-3X6.65X6.65X2.653X6.65X2.65X2.65-(2.65)^3=6*6*6
This is of the form
(a - b)^3 = a3 - 3a^2b 3ab^2 - b3
=(6.65)^3-3x(6.65)^2 x2.653(2.65)^2 x6.65-(2.65)^3
=(6.65-2.65)^3=4^3/6^3
=4x4x4/6x6x6 = 8/27
5.
(79.5 64.5 )^2(79.5-64.5)^2/(79.5)^2 (64.5)^2=
=2(a2b2)/a2b2=2
6.
Find (4.8)^3-0.027/(4.8)^21.440.09
This is of the form=a^3-b^3/a^2abb^2
a=4.8 and b=0.3
a^3-b^3/a^2abb^2=a-b
=4.8-0.3=4.5
7.
625*625*375*375-625*375/625*625*625375*375*375 =
This is of the form =a2b2-ab/a3b3
=1/ab
=1/625375
=1/1000=0.001
TYPE -IV

FRACTIONS

Any number of the form x/y is called a fraction.
TYPES
a.PROPER FRACTION:-
A fraction in which numerator is less than the denominator.
Eg:-1/5,2/3,etc
b.IMPROPER FRACTION:-
It is the one in which numerator is equal to or greater than denominator. It is also called vulgar fraction.
Eg:- 9/4,5/5
C.MIXED FRACTION:-
A figure consisting of a whole number and a proper fraction.
Eg-3 2/5,4 6/8
(A) BASIC PROPERTIES

*The value of a fraction is not changed if both the numerator and denominator are multiplied by the same number.
=>a/b=axc/ bxc
*The value of a fraction is not changed if both the numerator and denominator are divided by the same number.
=>a/b=a/c / b/c
*A fraction can be simplified to lower terms by dividing both numerator and denominator with their common factors until further division is not possible.
Eg:-9/27=9/3/27/3=3/9=3/9=3/3/9/3=1/3
POINTS TO REMEMBER

*The value of Roper fraction is always less than '1'.
*The value of Improper fraction is always more than or equal to '1'
*An improper fra-ction can be converted to a mixed fraction and vice versa
Eg:-8 2/4=(8*4)2/4
=34/4
(B) ADDITION AND SUBTRACTION OF FRACTIONS

1.If the denominators of the fractions are same, just add or subtract the numerator.
Eg:- 2/64/6=6/6=1
2.If the denominators of the fractions are not same, LCM of the denominators are found to make it same and then solve as follows:-
Eg:-(1) 2/63/12
1st => LCM of 6 and 12 is 12.
So 12 is the denominator.
Then, 12 / 6 = 2 and 2 x 2 = 4
12 /12 = 1 and 1 x 3 = 3
(2)2/6X12 3/12xl2/2
.'. 43/12 = 7/12
(3)3/155/7
1st => LCM of 15 and 7 is 105.
So denominator is 105.
Then, 105 / 15 = 7 and 7 x 3 = 21
105 / 7= 15 and 15x5 = 75
.'.75 21 /105 = 96/105 =32/35
(C) MULTIPLICATION OF FRACTIONS

1.If a fraction is multiplied by a whole number multiply the numerator by the whole number.
Eg:- 5x6/7=5*6/7=30/7
2.If a fraction is multiplied by another fraction, multiply corresponding numerators and denominators.
Eg:-3/5* 4/7=3x4/5*7= 12 /35
(D)DIVISION OF FRACTION

1.To divide a fraction by a whole number, multiply the denominator by the whole number.
Eg:-2/5 / 7=2/5*7=2/35
2. find the a fraction by another Action, multiply the reciprocal °f the divisor and
Eg:-4/3 / 5/7 = 4/3*7/5 = 28/15

EXAMPLES

1.
Which fraction is the biggest.
6/13,9/13,4/13 and 11/13
If denominator is same, the one with highest value of numerator is the biggest
fraction. So here it is 11/13
2.
Which is the biggest among
9/4,9/2,9/11 and 9/16
If numerator is same, the one with the lowest denominator will be the biggest fraction.
So here it is 9/2
3.
Find the value of x in 5*5/6-3-8/9- x=1
35/6-35/9 x=1
105-70/18- x=1
35/18 - x =1
x=35/18-1
.'.x=17/18
4.
1/1=?
11/11/2
=1 1/11/2 =11/3/2
=12/3=5/3
5.
1/7 / 1/7 o f1/7 / 1/7 of 1/7 / 1/7=___
1/7 / 1/7 * 1/7 / 1/7 * 1/7 / 1/7 = 1/7 /1/49 / 1/49 / 1/7
1/7*49 / 1/49*7=7/1/7=49
6.
Find the number 1th/9 of which exceeds its 12th part by 100
Let the number be x
Then,1/9*1/12 x=100
4x-3x/36=100
.'.x =3600

(E) DECIMALS

Fractions that have powers of 10 ie, 10, 10^2, 10^3, etc in the denominators are called Decimal fractions.
Eg:- 0.6, 7.235, etc
0.6 = 6/10;7.235= 7235/1000

RECURRING DECIMALS

If a number or a set of numbers is repeated in a decimal fraction, it is called recurring decimal.
Eg:-1/3=0.333 = 0.3=0.3

PURE RECURRING DECIMALS

The decimal fractions in which all the numbers after the decimal point are repeated are called pure recurring decimal.
Eg:- (1) 0.333 = 0.3
(2) 0.636363 = 0.63

MIXED RECURRING DECIMALS

The decimal fractions in which some numbers after the decimal point are repeated and some numbers are not repeated are called Mixed Recurring Decimals.
Eg:- 0.75353............................. = 0.753 -

EXAMPLES

1.
Find 0.1728/1.2
=0.1728 (x10000) /1.2 (x10000)
=1728/12000=0.144
2.
Find the value of 43.63 ?
x = 43.63
x = 43.63 63 63 _____(i)
100x = 4363. 63 63 63 _______ (ii)
(ii) - (i) => 99x = 4320
.'.x=4320/99
3.
Find 0.39325/3.25
=0.39325 (x100000)/ 3.25 (x100000)
=39325/325000=0.121

PRACTICE EXERCISES

1.(469 174)^2 - (469 —174)^2/469x174
2.A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, find the number of hens.
3.A sum of Rs. 5100 has been divided
among A, B and C such that A gets 2rd/3 3
of what B gets and B gets 1th/4 of what C gets.
What is the share of B
4.(79-4)x(11-6)/(5x9)-(1312)
=0.68x0.680.6x0.6-2x0.68x0.6 /0.68-0.6= =
6.If (a - b) is 20 more than (cd) and (ab) is 10 less than (c - d),find a - c.
7.The product of two numbers is 32. The sum of their squares is 80. Find the sum of the two numbers.
8.Find the sum of the smallest 6 digit number and the greatest 5 digit number.
9.65x2 3/5 = ?x6 7/5
10.If the sum of two numbers exceeds their difference by 20, find the smaller of the two numbers.
11.The difference of two numbers is 10 and 1 th/5 of their sum is 14. Find the greater number.
12.Simplify 3.6*3.62.3x2.3-2x2.3x3.6
13.A man spends 1th/5 of his income for house rent,1th/8 for clothings and 1 th/10 for food. What is his total income if he saves 32200?
14.If 4 4x1-3=0,find value of x.
15.The sum of squares of two numbers is 468 and the square of their difference is 36. Find the product of the two numbers.

ANSWER WITH EXPLANATION

1.
This is of the form (ab)^2-(a-b)^2/ab
=4ab/ab=4
2.
Let the number of hens be x and the number of cows be y.
Then, x y = 48 ______(i)
2x 4y = 140
=> x 2y=70____(ii)
Solving (i) and(ii) =>
xy =48
-x-2y=-70
-y=-22
ie, y = 22
.'.x = 26
.'. Number of hens = 26
3.
Let C's share = Rs. x
Then, B's share = Rs.x/4
A's share = Rs.[2/3*x/4]=Rs.x/6
.'.x/6x/4x=5100
17 x/12=5100
x =5100*12/17=3600
.'.B's share = Rs. (3600/4)=Rs.900
4.
(79-4)(11-6) /(5x9)-(1312)= 75x5/ 45-25
=75x5/20= 75/4
5.
This is of the form a2b2-2ab/a-b
(a-b)2/a - b
=a-b
= 0.68 - 0.6 = 0.08
6.
(a-b) = 20 (cd) ——(i)
(a b) = c-d - 10 ——— (ii)
(i) (ii)=>
2a = 20 2c-10
2a= 2c 10
2a-2c=10
a - c=5
7.
Let the two numbers be x and y
.'.xy = 32
x2 y2 = 80
(xy)2 = x2 y22xy
= 80 2 x 32
= 80 64 = 144
.'. xy = 12
.'. Sum of the numbers = 12
8.
Smallest 6 digit number = 100000
Largest 5 digit number = 99999
sum = 100000 99999
= 199999
9.
65x2 3/5 = x *6 7/5
65*13/5 =x *37/5
.'. x =169*5/37=22.8
10.
x y=(x-y) 20
x y-x y= 20
2y = 20
.'.y = 10
.'. the smaller number is 10
11.
Let the numbers be x and y
Then, x-y =10 - _____(i)
x y/5=14
xy =70____(ii)
x-y =10 ____(i)
(i)&(ii)=>
2x=80
x=40
.'. Greater number
12.
3.6*3.62.3*2.3-2.3*3.6
This is of the form a2 - 2ab b2 =(a-b)2
.'. (a-b)2= a-b
=3.6-2.3 = 1.3
13.
Let the total income be x
Then, x =1/5x1/8x1/10x32200
x-1/5x-1/8x-1/10x=32200
40x -8x -5x-4x/40= 32200
23x/40= 32200
x=32200x40 x = 23
= 56,000
.'.Income = Rs. 56,000
4x 1-3=0 = 0
4x 1 = 3
4x 1 =3^4
4x 1 =81
4x = 80
.'. x=20
15.
Let the numbers be x and y.
Then, x2 y2 = 468
(x - y)2 = 36
(x - y)2 =x2y2-2xy
xy =x2 y2-(x-y)2/ 2
=468-36/2 =432/2
= 216
.'.Product of the two numbers =216

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