Things you should know in algebra before GRE

Componendo and dividendo:

If x/y = z/t then; (x+y)/y=(z+t)/t.

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

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

Equations and Exponents

• (x+y)² = x² + y² + 2xy
• (x-y)² = x² + y² - 2xy
• (x+y)³ = x³ + y³ + 3xy(x + y)
• (x-y)³ = x³ - y³ - 3xy(x - y)
• x² - y² = (x + y)(x - y)
• x^-2 = 1/x², 2^-4 = 1/16 = 1/2⁴
• (x^a)(x^b) = x^a+b
• x^a/x^b = x^a-b = 1/x^b-a
• x⁰=1
• x^ay^a = (xy)^a, 2²3² = 6²
• quadratic equation, for any given x if ax² + bx + c =0 then x has 2 solutions
  x=(-b+√(b² - 4ac)/2a, x=(-b-√(b² - 4ac)/2a
• x^ay^b is not equal to (xy)^a+b

Functions

f(x)= 9x + 5; where f is function of x (why? because it changes with x, it depends on x). value of f can be obtained by substituting x in '9x + 5', so f(2) =23 when x=2.
The domain of any function is set of all permissible inputs values of the variable (in above case it's all permissible values of x).
The Range is set of all values of y.

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 + b², then 5@2 = 3(5) +(2)²

INEQUALITIES:

The multiplication or division of an inequality by a negative number causes a reversal of the direction of the inequality sign.E.g., 5 - 4x > 7 becomes -5 + 4x < 7 Note: Do not forget to charge the inequality sign (‘<’ or ‘>’). "Reciprocal (or) inversion" also causes of direction of the inequality sign i.e., a/b > c/d becomes b/a < d /c. x2>x You Cannot divide both sides by x and say x>1. x2-x>0 x(x-1)>0 Solution would be either both x and x-1 are greater than zero, or both x and x-1 are smaller than zero. So, your solution is: x>1 or x<0

-3x + 5 < 23
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)

Averages

Average of a set of n numbers is the sum of those numbers divided by n, Average = (sum of n numbers)/n or Average = sum/n.

Arithmetic Sequence If n numbers form an arithmetic sequence wherein the difference between any 2 consecutive terms is the same, then the average of the numbers is the middle term in the sequence, if n is odd and if n is even , the average of the numbers is the average of the two middle terms, or it also is the average of the first and the last numbers.
Distance, Speed, Time and Rate

Distance = rate x time

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 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.

Distance, Time and Speed

Distance : Distance = rate x time
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.

Time and work:
1/(total Work_ = 1/(Work Rate1) + 1/ (Work Rate2)

Output = rate x time

Simple Interest
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)

Compound Interest
A = P (1+ r)ⁿ n indicates the number of compounding years. Compound interest 20% calculated semi - annually is 10% for 6 months.

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