x/(x+y) =z/(z+t)

(x-y)/y = (z-t)/t

x/(x-y) =z/(z-y)

(x+y)/(x-y) = (z+t)/(z-t).

- (x+y)
^{2}= x^{2}+ y^{2}+ 2xy - (x-y)
^{2}= x^{2}+ y^{2}- 2xy - (x+y)
^{3}= x^{3}+ y^{3}+ 3xy(x + y) - (x-y)
^{3}= x^{3}- y^{3}- 3xy(x - y) - x
^{2}- y^{2}= (x + y)(x - y) - x
^{-2}= 1/x^{2}, 2^{-4}= 1/16 = 1/2^{4} - (x
^{a})(x^{b}) = x^{a+b} - x
^{a}/x^{b}= x^{a-b}= 1/x^{b-a} - x
^{0}=1 - x
^{a}y^{a}= (xy)^{a}, 2^{2}3^{2}= 6^{2} - quadratic equation, for any given x if ax
^{2}+ bx + c =0 then x has 2 solutions

x=(-b+√(b^{2}- 4ac)/2a, x=(-b-√(b^{2}- 4ac)/2a - x
^{a}y^{b}is not equal to (xy)^{a+b}

The

The

functions questions can also be represented with strange operators (like a # b) and the question might be (x # 2y) : just replace ‘a’ and’ b’, with ‘x’ and ‘y’. for example: a@ b =3a + b

for above inequality we need solutions for x.. above equation is equivalent to -3x < 18 or -x < 6 which means x > -6

hence x can be -5,-3.. 1,2... (infinite values)

Distance, Speed, Time and Rate

If two objects with speeds x miles per hr and y miles per hr are moving in the same direction, then their relative speed is given by (x - y) miles/hr.

If they are moving in opposite direction, then their relative speed is given by (x + y) miles/hr.

The average speed is not the average of the two speeds in the problem.

rate = distance /time

time = distance /rate.

Word problem that involves speed, Time and distance, we should always note whether the problem situation involves, Motion in the "same direction" Motion in the "opposite direction" or "A round trip."

If two objects with speeds x miles/hr and y miles /hr are moving in the same direction, then their relative speed is given by (x - y) miles/hr.

If they are moving in opposite direction, then their relative speed is given by (x + y) miles/hr.

The average speed is not the average of the two speeds in the problem.

1/(total Work_ = 1/(Work Rate1) + 1/ (Work Rate2)

Output = rate x time

A = P (1+ r n) , A=amount to be earned or returned, P = amount invested or borrowed, n = time (no of years), r = ratio of interest (in percent)

A = P (1+ r)

Compound interest: final balance = principal x (1 + interest rate /100c) (time) (C). . .. C: no. of time compounded annually principal x (1 + 20 / (100 x 2)) time x 2

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