NguyenHaiHa

17-08-2006, 04:25 PM

During my preparation for GRE, I complied this stuff. These experiences were contributed by korean and chinese GRE aspirants who succeeded in GRE. However, learning Math formulae by heart is not the rule of thumb for GRE. For those who are new bees of GRE, the best advice is that you should read GRE Math Review of ETS in an effort to understand the range of math knowledge tested in GRE exams, familarize yourself with Math vocabulary and then brush up on the basic Math concepts

NguyenHaiHa

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) : (vm - v2)(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]

This formula can be applied to problem involving dilution of milk with water, etc...

EXPLAINATION TO THE ABOVE FORMULA

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) (I assume that you understood this... )

vm (n1 + n2) = v1 n1 + v2 n2

n1 (vm - v1) = n2 (v2 - vm)

so, n1/n2 = (v2 - vm)/(vm - v1) ----> (1)

similarly if you mix n2 and n3, then their ratio would be given by

n2/n3 = (v3 - vm)/(vm - v2) ----> (2)

now assume we mix n1, n2 and n3 of different ingredients of value v1, v2 and v3. the individual ratios (1) and (2) will still be the same.

now combine these ratios to get n1:n2:n3 by making the denominators common

n1/n2 = (v2 - vm)(v3 - vm)/(vm - v1)(v3 - vm) and

n2/n3 = (v3 - vm)(vm - v1)/(vm - v2)(vm - v1)

rearrange this and you will get the formula:

n1 : n2 : n3 = (v2 - vm)(v3 - vm) : (vm - v1)(v3 - vm) : (vm - v2)(vm - v1)

Hope this is clear...

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 consecutive 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 = ac

HARMONIC PROGRESSION

A series of non-zero numbers is said to be harmonic progression (abbreviated H.P.) if the series obtained by taking reciprocals of the corresponding terms of the given series is an arithmetic progression.

For example, the series 1 +1/4 +1/7 +1/10 +..... is an H.P. since the series obtained by taking reciprocals of its corresponding terms i.e. 1 +4 +7 +10 +... is an A.P.

A general H.P. is 1/a + 1/(a + d) + 1(a + 2d) + ...

nth term of an H.P. = 1/[a +(n -1)d]

Three numbers a, b, c are in H.P. iff 1/a, 1/b, 1/c are in A.P.

i.e. iff 1/a + 1/c = 2/b

i.e. iff b= 2ac/(a + c)

Thus the H.M. between a and b is H = 2ac/(a + c)

----------------------------------------------------------------------------------------

If A, G, H are arithmetic, geometric and harmonic means between two distinct, positive real numbers a and b, THEN

1. G² = AH i.e. A, G, H are in G.P.

2. A, G, H are in descending order of magnitude i.e. A > G > H.

NguyenHaiHa

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) : (vm - v2)(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]

This formula can be applied to problem involving dilution of milk with water, etc...

EXPLAINATION TO THE ABOVE FORMULA

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) (I assume that you understood this... )

vm (n1 + n2) = v1 n1 + v2 n2

n1 (vm - v1) = n2 (v2 - vm)

so, n1/n2 = (v2 - vm)/(vm - v1) ----> (1)

similarly if you mix n2 and n3, then their ratio would be given by

n2/n3 = (v3 - vm)/(vm - v2) ----> (2)

now assume we mix n1, n2 and n3 of different ingredients of value v1, v2 and v3. the individual ratios (1) and (2) will still be the same.

now combine these ratios to get n1:n2:n3 by making the denominators common

n1/n2 = (v2 - vm)(v3 - vm)/(vm - v1)(v3 - vm) and

n2/n3 = (v3 - vm)(vm - v1)/(vm - v2)(vm - v1)

rearrange this and you will get the formula:

n1 : n2 : n3 = (v2 - vm)(v3 - vm) : (vm - v1)(v3 - vm) : (vm - v2)(vm - v1)

Hope this is clear...

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 consecutive 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 = ac

HARMONIC PROGRESSION

A series of non-zero numbers is said to be harmonic progression (abbreviated H.P.) if the series obtained by taking reciprocals of the corresponding terms of the given series is an arithmetic progression.

For example, the series 1 +1/4 +1/7 +1/10 +..... is an H.P. since the series obtained by taking reciprocals of its corresponding terms i.e. 1 +4 +7 +10 +... is an A.P.

A general H.P. is 1/a + 1/(a + d) + 1(a + 2d) + ...

nth term of an H.P. = 1/[a +(n -1)d]

Three numbers a, b, c are in H.P. iff 1/a, 1/b, 1/c are in A.P.

i.e. iff 1/a + 1/c = 2/b

i.e. iff b= 2ac/(a + c)

Thus the H.M. between a and b is H = 2ac/(a + c)

----------------------------------------------------------------------------------------

If A, G, H are arithmetic, geometric and harmonic means between two distinct, positive real numbers a and b, THEN

1. G² = AH i.e. A, G, H are in G.P.

2. A, G, H are in descending order of magnitude i.e. A > G > H.