# Equation

#### Saumyojit

##### Full Member
IF a/b = b/c=c/d then solve (b^3 + c^3 + d^3)/(a^3+b^3+c^3)

I dont want an approach using k.

I have started like this :
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2

How will i minimize (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ???

#### Mr. Bland

##### Junior Member
What exactly is it that needs to be solved? a, b, c and d can be solved using only the proportions, and the other expression isn't an equation.

#### Saumyojit

##### Full Member
What exactly is it that needs to be solved? a, b, c and d can be solved using only the proportions, and the other expression isn't an equation.
It needs to be minimised to a/d .
How can I bring it without using k .

#### firemath

##### Full Member
I don't see a k.......

#### Saumyojit

##### Full Member
I don't see a k.......
most of the people would solve it using a/b = b/c=c/d = k
a=bk.....
I dont want to use k
i want to approach it like this
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2

#### firemath

##### Full Member
I see......I was taught that formula differently.
So you want a compressed answer in variable terms?

#### Saumyojit

##### Full Member
I see......I was taught that formula differently.
So you want a compressed answer in variable terms?
see i know that
ac=b^2

b/c=c/d => bd =c^2

how will i substitute
ac=b^2
bd =c^2 in (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ?

a=b^2/c
c=b^2/a
b=c^2/d
d=c^2/b

Do you follow?

#### Dr.Peterson

##### Elite Member
IF a/b = b/c=c/d then solve (b^3 + c^3 + d^3)/(a^3+b^3+c^3)

I dont want an approach using k.

I have started like this :
a/b = b/c =>
ac=b^2

b/c=c/d => bd =c^2

How will i minimize (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ???
Why not use the k method, which seems simplest?

I think you mean, not "minimize" (which means, find the lowest value of), but "simplify".

see i know that
ac=b^2

b/c=c/d => bd =c^2

how will i substitute
ac=b^2
bd =c^2 in (b^3 + c^3 + d^3)/(a^3+b^3+c^3) ?
One way is to express c and d in terms of a and b: c = b^2/a, d = c^2/b = b^3/a^2. Make those substitutions, and simplify.

Using a/b = b/c = c/d = k (or, as I prefer, b/a = c/b = d/c = k) is a lot less complicated.

#### Saumyojit

##### Full Member
Why not use the k method, which seems simplest?

I think you mean, not "minimize" (which means, find the lowest value of), but "simplify".

One way is to express c and d in terms of a and b: c = b^2/a, d = c^2/b = b^3/a^2. Make those substitutions, and simplify.

Using a/b = b/c = c/d = k (or, as I prefer, b/a = c/b = d/c = k) is a lot less complicated.
I have substituted c = b^2/a, d = c^2/b = b^3/a^2 .
but i am stuck after 2 steps and i cannot further minimize it .
Please show me the whole steps using this method : c = b^2/a, d = c^2/b = b^3/a^2

#### Saumyojit

##### Full Member
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) =(abc +bdc+d^3)/(a^3+abc+bdc)=then what?

b^3=abc

c^3=bdc

#### Dr.Peterson

##### Elite Member
That's not the substitution I suggested. Why did you say, "I have substituted c = b^2/a, d = c^2/b = b^3/a^2"? You didn't.

The idea is to express everything in terms of a and b only, so that you can simplify the expression. For example, c^3 will become (b^2/a)^3 = b^6/a^3, and d^3 will become (b^3/a^2)^3 = b^9/a^6. You'll have a complex fraction that you can simplify by multiplying numerator and denominator by the LCD.

#### Saumyojit

##### Full Member
USING UR METHOD :
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) = {b^3+b^6/a^3 +b^9/a^6 }/{a^3+b^3+b^6/a^3)}
=> {b^3 *a^6 +b^3*a^3+b^9}/a^6 * a^3 / {a^6+a^3 * b^3 + b^6} =>( b^3 *a^6 +b^3*a^3+b^9)/(a^6+a^3 * b^3+ b^6) * 1/a^3

after doing lcm

=>b^3 (a^6 +a^3+b^6)/a^3(a^6+a^3 * b^3+ b^6) then what??

#### Dr.Peterson

##### Elite Member
Then, since the parenthesized factors in numerator and denominator look so similar, you should check your work to see if maybe they are really identical and can be cancelled!

#### Saumyojit

##### Full Member
I have tried every possible way but not minimised

#### Dr.Peterson

##### Elite Member
Please do as I said, and check your work on factoring the numerator. Is it really true that

{b^3+b^6/a^3 +b^9/a^6 } = {b^3 *a^6 +b^3*a^3+b^9}/a^6 ?​

A major part of doing mathematics well is to learn to check your work and catch errors.

#### Subhotosh Khan

##### Super Moderator
Staff member
USING UR METHOD :
(b^3 + c^3 + d^3)/(a^3+b^3+c^3) = {b^3+b^6/a^3 +b^9/a^6 }/{a^3+b^3+b^6/a^3)}
=> {b^3 *a^6 +b^3*a^3+b^9}/a^6 * a^3 / {a^6+a^3 * b^3 + b^6} =>( b^3 *a^6 +b^3*a^3+b^9)/(a^6+a^3 * b^3+ b^6) * 1/a^3

after doing lcm

=>b^3 (a^6 +a^3+b^6)/a^3(a^6+a^3 * b^3+ b^6) then what??
Did you REVIEW your work and are you certain that above expression is correct?

#### Dr.Peterson

##### Elite Member
@Saumyojit: To encourage you to continue, I will add that there is just one number wrong, and after fixing it you will be almost finished.