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

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Senior Manager
Joined: 13 Jun 2013
Posts: 255
If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 05:36
1
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Difficulty:

85% (hard)

Question Stats:

58% (02:53) correct 42% (02:59) wrong based on 142 sessions

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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
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1727
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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21 Jan 2015, 21:39
4
1
$$\frac{b}{(a-c)} = \frac{a}{b}$$

$$b^2 = a^2 - ac$$ ............. (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

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

##### General Discussion
Manager
Joined: 31 Jul 2014
Posts: 125
GMAT 1: 630 Q48 V29
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 06:29
manpreetsingh86 wrote:
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

I got answer correct, but want to know other way to solve.
So Kindly provide OE
Senior Manager
Joined: 13 Jun 2013
Posts: 255
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 09:07
manpreetsingh86 wrote:
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

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

Manager
Joined: 31 Jul 2014
Posts: 125
GMAT 1: 630 Q48 V29
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 09:48
1
manpreetsingh86 wrote:
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

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.
Senior Manager
Joined: 13 Jun 2013
Posts: 255
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 11:28
2
$$\frac{b}{a-c} = \frac{a+b}{c} = \frac{a}{b} =k$$

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.
Intern
Joined: 09 May 2013
Posts: 15
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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16 Jan 2015, 11:55
manpreetsingh86 wrote:
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

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

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

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

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

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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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01 Feb 2017, 04:54
1
1
manpreetsingh86 wrote:
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

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

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.

_________________
Karishma
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VP
Joined: 07 Dec 2014
Posts: 1217
If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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01 Feb 2017, 21:07
manpreetsingh86 wrote:
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

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
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Posts: 13737
Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff  [#permalink]

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12 Jun 2019, 03:47
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Re: If b/(a-c)=(a+b)/c = a/b for three positive number a,b and c, all diff   [#permalink] 12 Jun 2019, 03:47
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