MathRevolution wrote:

[GMAT math practice question]

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If the distances between consecutive ticks in the above number line are the same, which of the following points represents \(2^{11}\)?

A. A

B. B

C. C

D. D

E. E

Find the distance between two known values. Divide by number of intervals between them. That = interval length. Thus:

\(\frac{Upper-Lower}{No.OfIntervals}=\) Interval length

\(\frac{2^{10}-2^9}{2intervals}=\frac{2^9(2^1-1)}{2}=\frac{2^9}{2}=\) interval length

How

many intervals between \(2^{11}\) and \(2^9\)?

Total distance between them, divided by interval length = number of intervals:

\(\frac{Upper-Lower}{IntervalLength}=\) Number of intervals

(Upper-Lower) = Distance between \(2^{11}\) and \(2^9\)

\((2^{11} - 2^9) = 2^9(2^2 - 1) =\) \(2^9(3)\)

Divide that distance by interval length to get number of intervals, x

\(\frac{2^9(3)}{\frac{2^9}{2}}=x\)

\(2^9(3)*\frac{2}{2^9}=x\)

\(3 * 2 = x\)

\(x = 6\)

\(2^{11}\) is 6 intervals away from \(2^9\)

That is D on the line.

Answer D

*Faster but dense:

Between \(2^{10}\) and \(2^9\) there are TWO intervals. TWO intervals = what length?

\((2^{10} - 2^9) = 2^9(2^1 - 1)= 2^9(1)\) =

\(2^9\) Distance between \(2^{11}\) and \(2^9\):\((2^{11}-2^9) = 2^9(2^2-1)=2^9(3)\)

There are 3 lengths of \(2^9\) between \(2^9\) and \(2^{11}\)

--Length \(2^9 =\) 2 intervals

--(Number of lengths) * (number of intervals for that length) = (Total # of intervals)

--3 * 2 = 6 intervals = point D