Lisa invested $5000 for her retirement 20 years ago by

Question

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

Professor · Answer

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   times. 2^7 x 5000 = 640,000. so its value cannot be less than 400,000.  

using formulae, however, it is (1.25)^20 x 5000.

2times · Answer

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 ?

Professor · Answer

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.

ps_dahiya · Answer

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.

prashrash · Answer

Hey ps_dahiya, thanks for this really cool method for solving such a complex number. This works out v v fast

2times · Answer

ps_dahiya.. I like your 2^10 better approximation than my 5^5

smaller error

kevincan · Answer

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.

lan583 · Answer

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

ps_dahiya · Answer

Multiply numerator and denominator by 2^20

Lisa invested $5000 for her retirement 20 years ago by

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