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Mixtures first....
1. when you mix different quantities (say n1 and n2) of A and B, with different strengths or values v1 and v2 then their mean value vm after mixing will be:
Vm = (v1.n1 + v2.n2) / (n1 + n2)
you can use this to find the final price of say two types of rice being mixed or final strength of acids of different concentration being mixed etc....
the ratio in which they have to be mixed in order to get a mean value of vm can be given as:
n1/n2 = (v2 - vm)/(vm - v1)
When three different ingredients are mixed then the ratio in which they have to be mixed in order to get a final strength of vm is:
n1 : n2 : n3 = (v2 - vm)(v3 - vm) : (vm - v1)(v3 - vm) : (v2 - vm)(vm - v1)
2. If from a vessel containing M units of mixtures of A & B, x units of the mixture is taken out & replaced by an equal amount of B only .And If this process of taking out & replacement by B is repeated n times , then after n operations,
Amount of A left/ Amount of A originally present = (1-x/M)^n
3. If the vessel contains M units of A only and from this x units of A is taken out and replaced by x units of B. if this process is repeated n times, then:
Amount of A left = M [(1 - x/M)^n]
Ths formula can be applied to problem involving dilution of milk with water, etc...
(All these formulae have been taken from a test prep book issued by a private coaching center)
Lets make this thread an archive of formulae. (I dont know if there already exists one in the forum. I havent gone through all previous mails)
Everybody make an attempt to post any formula or short cut that you have got. Be it a standard formula that everybody knows or a rare one. coz there are people in this forum who are from non-math background and they might find it useful. (even if you are not sure how a formula works, as long as you know the formula does work for all cases, post it). So, lets get the ball rolling...
P.S. for any clarifications regarding how a formula works, start a new thread and let this thread be exclusively for listing formulae. that way we can reduce the length of this subforum and enable those, who will probably read this as a last minute revision, to spend lesser time...
More formulae:
PROGRESSION:
Sum of first n natural numbers: 1 +2 +3 + .... + n = [n(n+1)]/2
Sum of first n odd numbers: 1 + 3 + 5 + .... upto n terms = n^2
Sum of first n even numbers: 2 + 4 + 6 + ... upto n terms = n(n+1)
ARITHMETIC PROGRESSION
nth term of an Arithmetic progression = a + (n-1)d
Sum of n terms in an AP = s = n/2 [2a + (n-1)d]
where, a is the first term and d is the common differnce.
If a, b and c are any three consequtive terms in an AP, then 2b = a + c
GEOMETRIC PROGRESSION
nth term of a GP is = a[r^(n-1)]
sum of n terms of a GP:
s = a [(r^n - 1)/(r-1)] if r > 1
s = a [(1 - r^n)/(r-1)] if r < 1]
sum of an infinite number of terms of a GP is
s(approx.) = a/ (1-r) if r <1
If a, b and c are any three consequtive terms in a GP, then b^2 = a + c
PS: Everybody!!! please contribute or this thread will never grow.
Problems on trains
a km/hr = (a* (5/18)) m/s
a m/s = (a* (18/5)) km/hr
Time taken by a train of length l metres to pass a pole or a standing man or a signal post is equal to the time taken by the train to cover l metres.
Time taken by a train of length l metres to pass a stationary object of length b metres is the time taken by the train to cover (l+b) metres.
Suppose 2 trains or 2 bodies are moving in the same direction at u m/s and v m/s, where u>v, then their relative speed = (u-v) m/s
Suppose 2 trains or 2 bodies are moving in the opposite direction at u m/s and v m/s, where u>v, then their relative speed = (u+v) m/s
If 2 trains of length a metres and b metres are moving in opposite directions at u m/s and v m/s, then the time taken by the trains to cross each other = (a+b)/(u+v) sec
If 2 trains of length a metres and b metres are moving in same directions at u m/s and v m/s, then the time taken by the faster train to cross the slower train = (a+b)/(u-v) sec
If 2 trains(or bodies) start at the same time from points A and B towards each other and after crossing they take a and b sec in reaching B and A respectively, then
(A's speed)B's speed) = (root(B):root(A))
What you leave behind is not what is engraved in stone monuments, but what is woven into the lives of others.
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