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Lisa invested $5000 for her retirement 20 years ago by

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kevincan
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Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 14:10
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Lisa invested $5000 for her retirement 20 years ago by buying stocks. Luckily, these stocks have risen in value at least 25% in each of the last 20 years. The minimum worth of these stock is closest to

(A) $90,000 (B) $400,000 (C) $900,000 (D) $4,000,0000 (E) $9,000,000



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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 18:50
kevincan wrote:
Lisa invested $5000 for her retirement 20 years ago by buying stocks. Luckily, these stocks have risen in value at least 25% in each of the last 20 years. The minimum worth of these stock is closest to

(A) $90,000 (B) $400,000 (C) $900,000 (D) $4,000,0000 (E) $9,000,000


i got B using "rule of 72" because the answer choices have wide gap.

the money will doubled in almost 72/25 years. so in 20 years the money will doubled approx. 7 [(20)/(72/25)= [(20x25)/72] times. 2^7 x 5000 = 640,000. so its value cannot be less than 400,000. [a small note: if r = greater than 20 and smaller than 5, rule of 72 is not advisable to apply......]

using formulae, however, it is (1.25)^20 x 5000.
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 21:11
My ans is B...thought its an approximation so not sure....

first, get rid of the 5000
so the ans choices become A) 18 B)80 C)180 D)800 E)1800

and as we divided both sideds by 5000, we just need to approximate the value of (1.25)^20

=(5^3 * 10^-2)^20

= (5^60 - 10^-40)

Now i looked for any exponent of 5 that is close to 10,100, 1000, ....
the smallest is 5^5 = 3125; approximating this to 3000

3000^12 * 10^-40
3^12 * 10^36 * 10^-40

= 3^12 * 10^-4
= 81^3 * 10^-4
= 531441 * 10^ -4 (this step was a real pain)

= 53.1441

but we made a downward approximation (we used 3000 instead of 3125 that is a 4% discount so to say); now this step is a true leap of faith :)

53 with the approximation we made...... so 80 seems like a possible option.... but 180 seems too large

though intially when I chose the option I just chose because of the logic given above, but then... 80/53 =approx 1.6 and 180/53 = approx 3.6; so it seems unlikely that a 4% reduction (125/3125) can cause a 260% decrease in value; therefore B

Professor, is the rule of 72 universally used; I have never heard of it?
I tried that rule by first dividing the equation by 5000; so basically we need to find value of 2^7 = 128 ; which is closer to 180; like you said; but why different answers?

Anyone else with a better way to solve ?
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 21:57
2times wrote:
Professor, is the rule of 72 universally used; I have never heard of it?


yes, it is universally accepted but it doesnot give you the exact value. i mentioned its limitations aabove. it only gives a close approx when 5=< r =<20.

for ex: if annual interest rate (r) is 10%, how many years does a sum of deposit take to be doubled?

n = 72/10 = 7.2 years.
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 22:21
B

Answer is 5000 * (5/4)^20

Now lets solve (5/4)^20
(5/4)^20 = (5^20)/(2^40)
= 5^20 * 2^20/2^60
= 10^20/2^60
= 10^20/10^18
= 10^2 = 100
Because 2^10 = 1024 we can approximate 2^60 = 1024^6 = 10^18 approx

so (5/4)^20 will be less than 100. Probably near to 90 or so.

So answer must be less than 500,000 and probably near to 450,000.
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 23:03
Hey ps_dahiya, thanks for this really cool method for solving such a complex number. This works out v v fast :)
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   15 Aug 2006, 23:15
ps_dahiya.. I like your 2^10 better approximation than my 5^5

smaller error
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   16 Aug 2006, 03:15
Very resourceful, ps_dahiya! OA=B. How long did it take you?

Also 1.25^20= ((5/4)^4)^4*(5/4)^4

(5/4)^4=625/256 between 2.4 and 2.5


We know sqrt(6) is very close to 2.4 or 2.5, so ((5/4)^4)^4 shouldn't be far from 36.

36*2.5=90 36*2.4= 86, so we can expect her to have at least $430,000 or $450,000.
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   16 Aug 2006, 13:46
Dahiya,

How did you arrive at the bold numbers. Thanks

Now lets solve (5/4)^20
(5/4)^20 = (5^20)/(2^40)
= 5^20 * 2^20/2^60
= 10^20/2^60
= 10^20/10^18
= 10^2 = 100
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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink] New post   16 Aug 2006, 14:23
lan583 wrote:
Dahiya,

How did you arrive at the bold numbers. Thanks

Now lets solve (5/4)^20
(5/4)^20 = (5^20)/(2^40)
= 5^20 * 2^20/2^60
= 10^20/2^60
= 10^20/10^18
= 10^2 = 100

Multiply numerator and denominator by 2^20



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Re: Lisa invested $5000 for her retirement 20 years ago by [#permalink]
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