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If the distances between consecutive ticks in the above number line ar

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Math Revolution GMAT Instructor
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If the distances between consecutive ticks in the above number line ar  [#permalink]

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New post 11 Dec 2017, 00:54
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A
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D
E

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Question Stats:

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[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

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If the distances between consecutive ticks in the above number line ar  [#permalink]

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New post 11 Dec 2017, 03:38
MathRevolution wrote:
[GMAT math practice question]

Attachment:
pic.png


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


as the tick marks are equally spaced so it is an AP series.

let \(2^9=a\) and distance between ticks be \(d\)

so \(2^{10}=a+2d =>2^{10}=2^9+2d\)

\(d=2^8\)

let \(T_n=2^{11}\)

\(T_n=a+(n-1)d =>2^{11}=2^9+(n-1)2^8\). divide both sides by \(2^8\)

\(2^3=2+n-1=> n=7\)

so the \(7th\) tick mark will be \(2^{11}\) which is \(D\)

Option D
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If the distances between consecutive ticks in the above number line ar  [#permalink]

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New post 11 Dec 2017, 11:49
1
MathRevolution wrote:
[GMAT math practice question]

Attachment:
pic.png


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
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Re: If the distances between consecutive ticks in the above number line ar  [#permalink]

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New post 12 Dec 2017, 23:37
=>

Let \(d\) be the distance between consecutive ticks. Then
\(2d = 2^{10} – 2^9 = 2*2^9 – 2^9 = 2^9\)
\(d=2^8.\)
Since
\(2^{11} – 2^{10} = 2^32^8 – 2^22^8 = 8(2^8)– 4(2^8)= 4(2^8)= 4d,\)
\(2^{11}\) is the fourth point from \(2^{10}\), which is D.

Therefore, the answer is D.
Answer : D
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Re: If the distances between consecutive ticks in the above number line ar &nbs [#permalink] 12 Dec 2017, 23:37
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