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 Post subject: GMAT Algebra Posted: Thu Aug 05, 2010 6:00 am Joined: Sun May 30, 2010 2:23 am
Posts: 498
Let d > c > b > a. If c is twice as far from a as it is from d, and b is twice as far from c as it is from a, then
(d – b) / (d – a) =
A. 2/9
B. 1/3
C. 2/3
D. 7/9
E. 3/2

The correct answer is D.
Geometrical and algebraic solutions are below.

Top Post subject: Re: math: algebra, number line Posted: Thu Aug 05, 2010 6:20 am Joined: Sun May 30, 2010 2:23 am
Posts: 498
Since the exercise mentions a number line, draw one. Since d > a, label the left end of the number line with d and the right end with a. The values of b and c will be between a and d:

Start with the relationship between c, a, and d. Since c is twice as far from a as it is from d,
the segment between a and d is divided into 3 pieces – two pieces between c and a and one piece between d and c. Each piece is 1/3 of d to a.

The relationship between b, c, and a says that b is twice as far from c as it is from a. This
means the segment between a and c is divided into 3 pieces.

The segment between c and a was 2/3 of the segment between d and a. Dividing 2/3 into 3 pieces makes each piece (2/3) / 3 = 2/9.

So d – b (the length of the segment between d and b) makes 7/9 of d – a (the length of the segment between d and a).

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Top Post subject: Re: math: algebra, number line Posted: Thu Aug 05, 2010 6:34 am Joined: Sun May 30, 2010 2:23 am
Posts: 498
This question can also be solved algebraically. It might seem more difficult but it can be fast if you are good at fractions and simplifying expressions.

"c is twice as far from a as it is from d" means that
|c – a| = 2|c – d|
but since we know that d > c > b > a we can avoid using absolute values:
c – a = 2(d – c)

In much the same way we transform "b is twice as far from c as it is from a" into algebra:
c – b = 2(b – a)

Remember, that the question asks us to find: (d – b) / (d – a).

From formula c – a = 2(d – c) we can find that
d = (3c – a) / 2

From formula c – b = 2(b – a) we can find that
b = (c + 2a) / 3

Let us plug both formulas into expression we need to calculate:
(d – b) / (d – a) = [(3c – a)/2 – (c + 2a)/3] / [(3c – a)/2 – a] =
= [(7/6) × (c – a)] / [(3/2) × (c – a)]] =
= (7/6) / (3/2) = 7/9

Top Post subject: Re: math: algebra, number line Posted: Sun Oct 02, 2011 4:53 pm Joined: Sun May 30, 2010 3:15 am
Posts: 424
I don't quite understand the problem. I don't understand why 6 is twice as far from 0 as it is from 9. How is "if c is twice as far from a as it is from d, and b is twice as far from c as it is from a" interpret in the equation?

Top Post subject: Re: math: algebra, number line Posted: Sun Oct 02, 2011 5:11 pm Joined: Sun May 30, 2010 2:23 am
Posts: 498

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