Lining up A to H, from M-GAMT CAT

[sunilpunjabi]

would be great.

The Problem
A, B, C, D, E, F, G, and H are all integers, listed in order of increasing size. When these numbers are arranged on a number line, the distance between any two consecutive numbers is constant. If G and H are equal to 5^12 and 5^13, respectively, what is the value of A?

The solution My comments/questions in blue italics

The number line looks like.

A--B--C--D--E--F--G--H

The distance from G to H is 5^12 - 5^12

The distance between and two consecutive points is constant, so the distance from A to G will be 6 times the distance from G to H or 6(5^13 - 5^12). We are trying to fild the value of A, so I'm not sure why this is important, but I follow the logic of how to calculate the A to G distance.

The value of A, therefore, will be equal to the value of G minus the distance from A to G. This is where I start to get lost.
1. How does (A to G)-G=A? It does not seem logical to me.
2. I'm missing something here. If the question says "listed in order of increasing size"... "distance between any two consecutive numbers is constant". Why does one just not just extrapolate, if H=5^13 and G=5^12 then shouldn’t A=5^6?

The solution continues to solve for A, but this mean nothing to me since I'm stuck at understand the 2 items above.
5^12 - 6(5^13 - 5^12) --> 5^12 - 6[5^12(5 - 1)] --> 5^12 - 6(5^12)(4) --> 5^12(1 - 24) = (-23)5^12

Hope my post makes sense; try not to laugh too much at probably obvious mistakes I’m making.
Anyone have an explanation or alternate method, to help me understand to logic/solution?

Sunil

[RonPurewal]

first off, here's a post i made on another forum about this problem, which is farly comprehensive. check it out:
http://www.beatthegmat.com/consecutive- ... t9838.html

now for a couple of your comments:

1. How does (A to G)-G=A? It does not seem logical to me.

try this: take a number line, and just put zero, a, and g on it:
-------- 0 --------- a ----------- g ---------------

note that 'a' is the distance between 0 and a, and 'g' is the distance from 0 to g.
so:
(0 to g) minus (a to g) should equal (0 to a)
therefore
(g) minus (a to g) should equal (a)

the same is true even when a, g, or both are to the left of zero; try it and you'll see.

2. I'm missing something here. If the question says "listed in order of increasing size"... "distance between any two consecutive numbers is constant". Why does one just not just extrapolate, if H=5^13 and G=5^12 then shouldn’t A=5^6?

because that's not keeping a constant distance between numbers.
analogy:
think about 10^2, 10^3, and 10^4, which are 100, 1000, and 10,000 respectively.
the distance between the first two is only 900, but the latter two are 9000 apart. nowhere close to the same distance.
same goes for the powers of five in this problem.
(incidentally, the numbers you're trying to use do form a geometric series - not an arithmetic series - but that's irrelevant to this problem)

hth