If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff

Question

If  \frac{b}{(a-c)}   = \frac{(a+b)}{c} = \frac{a}{b}  for three positive number a,b, and c , all different, then  \frac{a}{b}=

a) 1/2
b) 3/5
c) 2/3
d) 5/3
e) 2

PareshGmat · Accepted Answer

\frac{b}{(a-c)} = \frac{a}{b}

b^2 = a^2 - ac  ............. (1)

\frac{(a+b)}{c} = \frac{a}{b}

b^2 = ac - ab  .................. (2)

As LHS is same, equating RHS of (1) & (2)

a^2 - ac = ac - ab

a - c = c - b

a+b = 2c .............. (3)

Replacing value of a+b in the original equation

\frac{a}{b} =\frac{2}{1}

Answer = E

anupamadw · Answer

I got answer correct, but want to know other way to solve.
So Kindly provide OE

manpreetsingh86 · Answer

hi, please post your approach.

anupamadw · Answer

b/(a-c) = (a+b)/c bc=(a-c)(a+b) bc=a^2+ab-ac-bc 2bc = a^2+ab-ac ---1 b/(a-c) = a/b b^2=a^2-ac ----2 (a+b)/c=a/b ab+b^2=ac b^2=ac-ab -----3 From 2 and 3 a^2-ac = ac-ab a^2+ab=2ac Substitute back in (1) 2bc=a^2+ab-ac 2bc=2ac-ac 2bc=ac ( c is +ve number ) a/b=2 Ans E But this is not nice way, I know

Could you please provide the quicker way to solve.

manpreetsingh86 · Answer

\frac{b}{a-c} = \frac{a+b}{c} = \frac{a}{b} =k

so, here we have to find the value of k.

b= ak-ck ------------1)

a+b =ck ---------------2)

adding 1) and 2) we have

a+2b = ak

a(k-1)= 2b

a/b = 2/k-1

also as per the question a/b =k

thus we have k = 2/k-1 ( after this step we can also plug-in the answer choices to get the final answer).

or k^2-k-2=0
(k+1)(k-2)=0
k=-1 or k=2.

out of these values only k=2 is given in the option. hence answer must be E.

tarangchhabra · Answer

b/(a-c)=(a+b)/c=a/b 
(I) (II) (III)

Using equations (I) & (III)
 b^2 = a^2 - ac  …………….. (IV)

Using equations (I) & (II)
 bc= a^2-ac+ab-bc 
 2bc=  +ab

Using equation (IV)
 2bc= (b^2)+ab

Thus, 
Either, b=0 (This is not possible due to given equality in question)
Or,  2c = a + b 
 2=(a+b)/c 
Using equality given in question
 a/b=2

I hope it is a simple solution

KarishmaB · Answer

\frac{b}{(a-c)}   = \frac{(a+b)}{c} = \frac{a}{b}

The value of a/b is something from the 5 options.

Note that  \frac{b}{(a-c)}   = \frac{a}{b} 
b * b = a * (a - c)
a, b and c are all positive numbers. (a - c) will be less than a so b must be less than a.
So only two options are possible - 5/3 or 2/1

Try 2/1 since it is easier.
If b = 2, a = 4. Then a - c = 4 - 3 = 1

Plug these values in here:  \frac{(a+b)}{c} , you get 6/3 = 2/1

It fits. This is the answer.

Answer (E)

gracie · Answer

equation1: b/(a-c)=a/b➡ac=a^2-b^2
equation2: (a+b)/c=a/b➡ac=ab+b^2
thus, a^2-b^2=ab+b^2➡
a^2-ab-2b^2=0➡
(a+b)(a-2b)=0
a+b=0 won't work as both numbers are positive
a-2b=0➡a=2b
only possible choice is a=2 and b=1
e

Kinshook · Answer

If  \frac{b}{(a-c)}   = \frac{(a+b)}{c} = \frac{a}{b}  for three positive number a,b, and c , all different, then  \frac{a}{b}=

Let  \frac{b}{a-c} = \frac{a+b}{c} = \frac{a}{b} = k

1. a = kb

2. b = ka - kc = k^2b - kc
(k^2 - 1)b = kc
c = (k^2- 1)b/k

3. a+b = kc
kb + b = (k^2-1)b
k + 1 = k^2 - 1
k^2 -k - 2 = 0
(k-2)(k+1) = 0
k = 2; since a, b & c are positive integers

\frac{a}{b} = k = 2

IMO E

If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff

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If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards