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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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11 Dec 2017, 00:54
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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
[Reveal] Spoiler: OA

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

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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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11 Dec 2017, 11:49
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.

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

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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.
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Re: If the distances between consecutive ticks in the above number line ar   [#permalink] 12 Dec 2017, 23:37
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# If the distances between consecutive ticks in the above number line ar

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