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# Aliens who live 1,000,000 miles from Earth decide to come to our plane

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Math Expert
Joined: 02 Sep 2009
Posts: 61544
Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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01 Dec 2019, 23:16
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Difficulty:

85% (hard)

Question Stats:

52% (03:19) correct 48% (03:37) wrong based on 52 sessions

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Competition Mode Question

Aliens who live 1,000,000 miles from Earth decide to come to our planet for a visit. They program their space shuttle to travel 100,000 miles per hour for the first 500,000 miles, 200,000 miles per hour for the next 250,000 miles, 400,000 miles for the next 125,000 miles and so on, doubling the speed every time half of the remaining distance is covered. If this pattern is maintained until the aliens reach earth, what will their average speed for the trip be in miles per hour?

A. 120,000
B. 125,000
C. 135,000
D. 150,000
E. 162,000

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Director
Joined: 25 Jul 2018
Posts: 578
Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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02 Dec 2019, 17:51
2
3
$$Speed(ave.)= (D_1+D_2+...D_n)/(T_1+T_2+...+T_n)$$

$$D_1=500000 ml$$
$$Speed_1=100000 mph$$
--> $$Time_1= \frac{500000}{100000}=5 h$$

$$D_2=250000$$ ml
$$Speed_2=200000$$ mph
--> $$Time_2=\frac{5}{4}$$

$$D_3=125000$$ ml
$$Speed_3= 400000$$ mph
--> $$Time_3= \frac{5}{16}$$

As you see the Distance and Time, they are creating the Infinite terms of Geometric Progressions series ($$S_∞=\frac{b}{(1-q)}$$ )
--> b is the first term, q is the common ratio ($$\frac{b_n}{b_{n-1}}$$)

Distance:

b= 500000
$$q = \frac{b_2}{b_1}= \frac{250000}{500000}= \frac{1}{2}$$
--> $$S_∞= \frac{500000}{(1-1/2)}= 1000000$$

Time:

b=5
$$q= \frac{b_2}{b_1}= \frac{1.25}{5}= \frac{1}{4}$$
--> $$S_∞= \frac{5}{(1-1/4)}= \frac{20}{3}$$

$$Speed(ave.)= \frac{Distance}{Time}= \frac{1000000}{(20/3)}= 150000$$

##### General Discussion
Director
Joined: 30 Sep 2017
Posts: 696
GMAT 1: 720 Q49 V40
GPA: 3.8
Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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02 Dec 2019, 08:08
2
Timing for each distance:
D1=500,000 miles, v1= 100mph and t1=D1/v1= 5 hours
D2=250,000 miles, v2= 200mph and t2=D2/v2= 1.25 hours
D3=125,000 miles, v3= 400mph and t3=D3/v3= 0.3125 hours, et cetera.

D (Total Distance) = 1,000,000 miles.
T (Total Time) = 5 + 1.25 + 0.3125 +... = 5/(1-0.25) = 20/3 hours.
Thus, v (average speed) = D / T = 150,000 mph

VP
Joined: 24 Nov 2016
Posts: 1247
Location: United States
Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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02 Dec 2019, 04:51
1
1
Quote:
Aliens who live 1,000,000 miles from Earth decide to come to our planet for a visit. They program their space shuttle to travel 100,000 miles per hour for the first 500,000 miles, 200,000 miles per hour for the next 250,000 miles, 400,000 miles for the next 125,000 miles and so on, doubling the speed every time half of the remaining distance is covered. If this pattern is maintained until the aliens reach earth, what will their average speed for the trip be in miles per hour?

A. 120,000
B. 125,000
C. 135,000
D. 150,000
E. 162,000

rate*time=distance
d1=500…r1=100…t1=500/100=5
d2=250…r2=200…t2=250/200=5/4
d3=125…r3=400…t3=125/400=25/80=5/16
d4=75…r4=800…t4=75/800=15/160=3/32
d5 to dn = 5%d… r5 to rn > 1600… t5 to tn < 50/1600 = 5/160 = 1/32

t=5+5/4+5/16+3/32+1/32=5+5/4+14/32 < 6.25 + 0.5 ~ 6.666
d=1000

rate=distance/time=1000/6.666=~150mph

Ans (D)
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Joined: 18 May 2019
Posts: 715
Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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02 Dec 2019, 19:12
1
Average Speed = Distance / Time
we need to determine the total distance.
Time to reach earth = 500,000/100,000 + 250,000/200,000 + 125,000/400,000 + 62,500/800,000 + 31,250/1,600,000 +...
=5 + 5/4 + 5/16 + 5/64 + ...
=5(1+1/4+1/16+1/64+...)
So we can look at this as an infinite geometric series.
common ratio = 1/4
so S=S1/(1-r) = 5/(1-1/4) = 5/(3/4) = 20/3
Hence Time taken = 20/3

Average Speed = 1,000,000/(20/3)= 3*50,000 = 150,000mph

Senior Manager
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Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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02 Dec 2019, 20:27
1
let d=1,000,000 miles, s=100,000 miles per hour
average speed= total distance/total time
= d/(d/2s)+(d/8s)+......= d/[d/s(1/2+1/8+1/32........)]
here denominator is a infinite GP series...whose sum = a/1-r
= (1/2)/(1-1/4) = 2/3
average speed= 3s/2=150,000 miles per hour

OA:D
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Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane  [#permalink]

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03 Dec 2019, 08:03
Given:
1. Aliens who live 1,000,000 miles from Earth decide to come to our planet for a visit. They program their space shuttle to travel 100,000 miles per hour for the first 500,000 miles, 200,000 miles per hour for the next 250,000 miles, 400,000 miles for the next 125,000 miles and so on, doubling the speed every time half of the remaining distance is covered.

Asked: If this pattern is maintained until the aliens reach earth, what will their average speed for the trip be in miles per hour?

Total distance travelled = 1,000,000 miles

Total time taken = 5 + 5/4 + 5/4^2 + .... = 5/(1-1/4) = 5*4/3 = 20/3 hrs

Average speed = 1000,000 miles / (20/3) hrs = 3000,000/20 = 150,000 miles/hrs

IMO D
Re: Aliens who live 1,000,000 miles from Earth decide to come to our plane   [#permalink] 03 Dec 2019, 08:03
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