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 Post subject: GMAT Algebra
PostPosted: Thu Aug 05, 2010 6:00 am 
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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.


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 Post subject: Re: math: algebra, number line
PostPosted: Thu Aug 05, 2010 6:20 am 
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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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 Post subject: Re: math: algebra, number line
PostPosted: Thu Aug 05, 2010 6:34 am 
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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


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 Post subject: Re: math: algebra, number line
PostPosted: Sun Oct 02, 2011 4:53 pm 
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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?


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 Post subject: Re: math: algebra, number line
PostPosted: Sun Oct 02, 2011 5:11 pm 
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