After investing some time in the preparation, it's not uncommon to realize that it would be better if we make our own notes which we can refer while revising these topics before the exam. In order to make this procedure simple, I have come up with this post.
1) Ratios and proportions:
If a certain amount N is divided among A, B in a ratio a:b then
A's share is a/(a+b) N and B's share is b/(a+b) N
If A : B = a:b and B:C = c:d then A:C = a c : bd
If A: B = p: q, B:C = r:s, C: D = t:u then
A: D = A/D = (A/ B ) (B/ C) (C/D)
If a quantity is increased by a/b then the new content becomes (a+b)/b
If a quantity is decreased by a/b then the new content becomes (b-a)/b
To reduce your computations, it is always recommended to remember these values in terms of percentages
1/1 = 100 %
1/2 = 50 %
1/3 = 33.33 %
1/4 = 25 %
1/5 = 20 %
1/6 = 16.66 %
1/7 = 14.28 % 1/13 = 7.6%
1/8 = 12.25 % 1/12 = 8.33%
1/9 = 11.11 % 1/11 = 9.09%
1/10 = 10 %
Proportion is a number considered in comparative relation to a whole
when 3 numbers are given a, b, c and asked to find the fourth proportion (x) then a/b = c/x
Mean proportion of two numbers a,b is x then x2 = ab
If A=BC then B, C are Inversely proportional and A,B or A,C are directly proportional
If A is kept constant, Increase in B should reflect decrease in A and vise versa
If B is kept constant, Increase in one term should reflect increase in another term.
log ab = log a + log b
log a/b = log a - log b
log(ab) = blog(a)
logab = 1/logba
logbn = logen/logeb
log10n = logen/loge10 = logen (0.43429448..)
The Highest power of a prime number 'P' (P is prime) in factorial of another number 'N' is
N/P + N /P2 + N/P3 + .....
The Highest power of a number 'Pa' where P is a prime number in N! is
[ N/P + N/P2 + N/P3 .......]/a
4)problems on Trains:
A Kilometer/Hour = A(5//18) meter/second.
A meter/second = A(18/5) Kilometer/Hour.
Time taken by a train of length 'l' units to cross a fixed point or pole is equal to the time taken by the train to travel 'l' units.
Time taken by a train of length 'l' units to cross a stationary object of length 'm' units is equal to the time taken by the train to travel 'l+m' units.
Suppose 2 trains are moving in the same direction with speeds U, V where U > V then their relative speed is U - V.
Suppose 2 trains are moving in opposite directions with speeds U, V then their relative speed is U + V.
If two trains start at same time from points A, B respectively towards each other and after crossing they take a and b sec to reach B, A respectively then
(A's speed): (B's speed) = (√b : √a)
( relative speed problems on trains )
1) Suppose 2 trains of lengths a, b are moving with speeds u, v (U>V)respectively then the time taken to cross each other is _________
Distance to cover = length of train a + length of train b = a+b .
Relative speed = (U - V)
Time=(a+b)/(U - V).
Relative speed = (U + V)
Time=(a+b)/(U + V).
5)Mixtures and allegations:
suppose a container contains x units of liquid out of which y units are replaced then the quantity of liquid after n such operations left in the container would be x(1-y/x)n
Quantity of cheap ingredient(b) : Quantity of costly ingredient (a)= x2-x1 : x1-x0
Simple Interest= (Profit Rate of Interest Time)/100
7)Profit - Loss - Discount:
Cost Price(C.P): Amount incurred for purchasing/manufacturing the product.
Selling Price(S.P): Gained amount in process of selling the product.
Actual price:Same as Maximum Retail Price.
Discount: price deducted from cost price.
Profit= S.P - C.P.
Loss= C.p - S.P.
Discount= Actual price - S.P.
profit % =(profit/C.P)100
loss % =(loss/C.P)100
100S.P = (100- Loss% )C.P or (100+Profit% )C.P
As discount is always calculated in customer's perspective, it deals with Actual price but not C.P
Discount % = (discount/Actual price)100.
when a trader sells the items at C.P but uses false weights then
profit % =[ Error/(True value-Error)]*100
8)Pipes and cisterns:
If pipe can fill the tank in 'x' hours then rate of flow is 1/x(1/x portion of tank gets filled in 1 hour)(when we divide the tank into x parts, one part gets filled in a unit of time)
If pipe can empty the tank in 'y' hours then emptying rate is 1/y(1/y portion of tank gets emptied in 1 hour)
The rate at which tank is filling =rate of inflow - the rate of outflow.
If a pipe fills the tank at 1/x rate and another pipe empties the tank in 1/y rate then on opening both the pipes:
If (x<y) rate at which tank is is 1/x-1/y
If (x>y) rate at which tank is emptying is 1/y-1/x
9)H.C.F and L.C.M:
LCM*HCF=product of the numbers.
If any number is the factor of other then LCM is the bigger one and HCF is the other.
10) Rivers boats and streams:
In water, the direction of the flow is downstream and opposite direction is upstream.
The speed of a boat in still water is u and speed of the stream is v then:
If speed upstream is a and speed downstream is b then:
speed in still water is (a+b)/2
the rate of flow of the stream is (a-b)/2