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# If d = (c - b)/(a - b), then b =

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Math Expert
Joined: 02 Sep 2009
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If d = (c - b)/(a - b), then b =  [#permalink]

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15 Mar 2018, 23:03
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84% (01:38) correct 16% (01:32) wrong based on 61 sessions

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If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

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If d = (c - b)/(a - b), then b =  [#permalink]

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16 Mar 2018, 01:07
1
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

We are given that d = $$\frac{c-b}{a-b}$$

Cross multiplying, da - db = c - b

$$db - b = c - ad$$ -> $$b(d - 1) = c - ad$$ -> $$b = \frac{(c - ad)}{(d - 1)}$$

Therefore, b = $$\frac{(c - ad)}{(d - 1)}$$ (Option D)

Alternate approach

If c=4,b=1,a=2 then d=4-1/2-1 = 3

Now substutiting in the available answer options,

A. (c - d)/(a - d) = (4-3)/(2-3) = -ive

B. (c + d)/(a + d) = (4+3)/(2+3) = 7/5

C. (ca - d)/(ca + d) = (8-3)/(8+3) = 5/11

D. (c - ad)/(1 - d) = (4-6)/(1-3) = -2/-2 = 1

E. (c + ad)/(d -1) = (4+6)/2 = 10/2 = 5 (Option D)
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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15 Mar 2018, 23:04
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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16 Mar 2018, 00:53
Bunuel wrote:
If d = (c - a)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

d = (c - a)/(a - b)

i.e. (a-b) = (c-a)/d

i.e. b = a - (c-a)/d

i.e. b = [ad - (c-a)]/d

Bunuel Are the options correct???
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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16 Mar 2018, 01:03
GMATinsight wrote:
Bunuel wrote:
If d = (c - a)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

d = (c - a)/(a - b)

i.e. (a-b) = (c-a)/d

i.e. b = a - (c-a)/d

i.e. b = [ad - (c-a)]/d

Bunuel Are the options correct???

There was a typo: it's d = (c - b)/(a - b) NOT d = (c - a)/(a - b). Edited. Thank you for noticing!
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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16 Mar 2018, 01:09
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

d = (c - b)/(a - b)

i.e. da - bd = c-b

i.e. bd - b = da - c

i.e. b (d-1) = da-c

i.e. b = (da - c) / (d-1) or

b = (c - ad)/(1 - d)

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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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19 Mar 2018, 16:14
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

First, we multiply both sides by (a - b) and expand the left side:

d(a - b) = c - b

da - db = c - b

Now, since our goal is to solve for b, we put all terms containing b on the left side of the equation and all other terms on the right side:

b - db = c - da

On the left side of the equation, factor out the common factor b from both terms:

b(1 - d) = c - da

Divide both sides by (1 - d), and now b is by itself on the left side of the equation:

b = (c - da)/(1 - d)

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If d = (c - b)/(a - b), then b =  [#permalink]

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19 Mar 2018, 19:32
pushpitkc wrote:
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

We are given that d = $$\frac{c-b}{a-b}$$

Cross multiplying, da - db = c - b

$$db - b = c - ad$$ -> $$b(d - 1) = c - ad$$ -> $$b = \frac{(c - ad)}{(d - 1)}$$

Therefore, b = $$\frac{(c - ad)}{(d - 1)}$$ (Option D)

Alternate approach

If c=4,b=1,a=2 then d=4-1/2-1 = 3

Now substutiting in the available answer options,

A. (c - d)/(a - d) = (4-3)/(2-3) = -ive

B. (c + d)/(a + d) = (4+3)/(2+3) = 7/5

C. (ca - d)/(ca + d) = (8-3)/(8+3) = 5/11

D. (c - ad)/(1 - d) = (4-6)/(1-3) = -2/-2 = 1

E. (c + ad)/(d -1) = (4+6)/2 = 10/2 = 5 (Option D)

I understand the algebraic approach, but I can not get alternative approach. Using your variables I arrived at the correct answer, but when I use the variables
a=1 , b=2, c=3, and d=-1, I get answer A and D equal to 2. Why can I not use this set of variables?
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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20 Mar 2018, 00:52
ScottTargetTestPrep wrote:
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

First, we multiply both sides by (a - b) and expand the left side:

d(a - b) = c - b

da - db = c - b

Now, since our goal is to solve for b, we put all terms containing b on the left side of the equation and all other terms on the right side:

b - db = c - da

On the left side of the equation, factor out the common factor b from both terms:

b(1 - d) = c - da

Divide both sides by (1 - d), and now b is by itself on the left side of the equation:

b = (c - da)/(1 - d)

I can follow your manipulations once I see the solution but how did you know - in two minutes - those were the right manipulations to get to the answer choices?..

I got stuck with b=c-da+db (which is correct but does not correspond to any answer choices...).
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If d = (c - b)/(a - b), then b =  [#permalink]

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20 Mar 2018, 10:25
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

$$d$$ = $$\frac{(c-b)}{(a-b)}$$ => $$d(a - b) = c - b$$ => $$ad - bd = c - b$$ =>
$$b - bd = c - ad$$ => $$b(1 - d) = c - ad$$ => $$b$$ = $$\frac{(c-ad)}{(1-d)}$$

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GMAT 1: 710 Q50 V36
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Re: If d = (c - b)/(a - b), then b =  [#permalink]

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20 Mar 2018, 10:27
grgsky wrote:
ScottTargetTestPrep wrote:
Bunuel wrote:
If d = (c - b)/(a - b), then b =

A. (c - d)/(a - d)

B. (c + d)/(a + d)

C. (ca - d)/(ca + d)

D. (c - ad)/(1 - d)

First, we multiply both sides by (a - b) and expand the left side:

d(a - b) = c - b

da - db = c - b

Now, since our goal is to solve for b, we put all terms containing b on the left side of the equation and all other terms on the right side:

b - db = c - da

On the left side of the equation, factor out the common factor b from both terms:

b(1 - d) = c - da

Divide both sides by (1 - d), and now b is by itself on the left side of the equation:

b = (c - da)/(1 - d)

I can follow your manipulations once I see the solution but how did you know - in two minutes - those were the right manipulations to get to the answer choices?..

I got stuck with b=c-da+db (which is correct but does not correspond to any answer choices...).

hi, ideally in these kind of questions, you'll want to concentrate all the value that related to b on one side of the equation and the rest on the other side. Check my solution above for details
Re: If d = (c - b)/(a - b), then b =   [#permalink] 20 Mar 2018, 10:27
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