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Let the loser's votes be = x
Then the winner's votes = x+220
Total votes = x + (x+220) =810
that is
x + x+ 220 = 810
2x+220 = 810
2x = 810 -220
2x = 590
x = 590/2
x= 295
x+220 = 295+ 220 = 515
Therefore the loser gets 295 votes
and the winner gets 515 votes

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Hypotenuse = 13ft
Length of one leg = 5ft
Let length of other leg be x ft
According to Pythogora's Theorem
Side^2 + Side^2 = Hypotenuse^2
5^2 + x^2 = 13^2
25 +x^2 = 169
x^2 =169 - 25
= 144
x= 12 ft
There fore the length of the other leg = 12 ft

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Let no: of adults be x
Cost of tickets for x adults = 12x
Let no: of children be y
Cost of tickets for y children = 4y
Total no: of tickets sold = x + y = 130 ----(1)
Total amount received =12x + 4y = 840 ----(2)
Equation (2) can be reduced to
3x + y = 210 ----(3) (this is obtained by dividing the equation with 4)
Subracting (1) from (3) we get
2x = 80
x = 40
No: of adults = 85
x + y = 130
40 + y = 130
y = 130 - 40 = 90
No: of children = 90

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Solution: Let the width of the original rectangle be x
Then length = 2x
Area=length x breadth =2x X x=2x^2
After the increase : width = x + 4
Length = 2x + 4
Area =(x + 4) X (2x + 4)
= 2x^2 + 12x + 16
This area is 88 SqMetres more than the previous area
ie. 2x^2 + 12x + 16 = 2x^2 +88
===> 12x + 16 = 88
===> 12x = 88 – 16
===> 12x = 72
===> x = 72/12
===> x = 6
therefore width= 6m and length = 2x = 2 X 6 = 12m

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Given 10x^2 - 3x - 4 = 0
10x^2 -8x + 5x - 4 = 0
2x(5x - 4) + 1(5x - 4) = 0
(5x-4)(2x+1) = 0
(I have written -3x as (-8x+5x) then took 2x common from the first two terms and 1 common for the next two terms . In the next step I took 5x-4 as common)

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A
\
\
\
\
\
C ______________\______________ D
O
In the above illustration angle AOC and angle AOD form a linear pair.
The pair is said to be linear because the sum of the angles is 180 degrees.
You can tell that their sum is 180degrees as the points C , O and D are collinear ie they lie in the same straight line. So angle COD = 180 degrees.
Therefore angle AOC + angle AOD = 180 degrees. You can say that angle AOC and angle AOD are adjacent angles . all linear pairs can be called adjacent angles

but all adjacent angles need not be linear pairs. It is not necessary that the sum of two adjacent angles to be equal to 180 degrees.

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Given: x + y =-1 ----(1)
1/x+1/y=1/2 -----(2)
1/x+1/y=1/2 ===> (x+y)/xy =1/2
x+y = xy/2 (x+y=-1 according to Equation 1)
-1 = xy/2 (Substituting x+y with -1 )
xy= -2
We know that (x-y)^2 = (x+y)^2-4xy
= (-1)^2 -4(-2)
= 1 -(-8)=1+8=9
(x-y)^2 = 9
therefore x - y = 3 ------(3)
x + y = -1 ------(1)
adding (3)&(1) we get 2x = 2
x = 1
x + y = -1
1 + y = -1
y = -1 -1 = -2
therefore x=1 , y= -2

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let the width be x yards
length=3x-75
perimeter=2(length+width)=2(x+3x-75)=
2(4x-75) = 370
4x-75 =370/2
4x-75 =185
4x=185+75
4x=260
x=260/4=65
therefore width=65yards , length= 3x-75=3x65-75=195-75=120yards

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solution:
(3y^2)(2y^3)=3x2x(y^(2+3))=6y^5
or
3xyxyx2xyxyxy=3x2xyxyxyxyxy=6y^5

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solution:
No
when two lines intersect ,then two pairs of vertically opposite angles are formed
so 20 and 20 is one pair of vertically opposite angles
if the third angle is 70 then the fourth angle should be 70
the sum of the angles=20+20+70+70=180
but the total of angles should be 360
so it is impossible for two lines to intersect in such a way that three of the angles are 20,70,20

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Solution:
8^(1/3) =
(2^3)^(1/3)= 2^(3x1/3)=2^1=2

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solution:
No: of adults = a
No: of children =c
children = 4 times adults
c = 4xa=4a
total=65
a+c=65
a+4a=65
5a=65
a=65/5=13
c=4a=4x13=52

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Solution :
(-3a^2b^-2)^2 / (5a^3)^2=
(-3^2 x a^4 x b^-4)/(5^2 x a^6) =
(-3 x -3 x a^4 x b^-4))/(5 x 5 x a^6) =
9a^4b^-4/25a^6 = (9a^-2b^-4)/25

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let us assume that it takes xhrs to return at 250mi/hr
then time taken to fly to St.Paul = (x+1)hrs at 200m/hr
distance=time x speed
onward distance= 250x
return distance=200(x+1)
onward distance=return distance
250x=200(x+1)
250x=200x+200
250-200x=200
50x=200
x=200/50
x=4hrs
distance=250x=250x4=1000miles

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It's called a reflex angle

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(x-2)(3x-6)(x+3)=
(x-2)[3(x-2)](x+3)=3[(x-2)^2](x+3)= 3(x^2-4x+4)(x+3)=
=3(x^3-3x^2-8x+12)

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let us solve the numerator first:
Numerator =4(-3/5)-3(1/4)= -12/5 -3/4 =(-12x4-3x5)/20 = (-48-15)/20= -63/20
Denominator =[2(-3/5)]= -6/5
Numerator/denominator =(-63/20)/(-6/5)=(-63/20)x(-5/6)=315/120=21/8