short cut!!

[Weekender]

The price of a microchip declines by 66% every 6 months. At this rate, approximately how many yrs will it take for the price of an $81 microchip to reach $1 per chip?

i got the ans as 2yrs.....but my method was too long
just want to know the shortest way possible

[arjmen]

66% ~ 2/3

81*[1-(2/3)]^n = 1
=> (1/3)^n = 1/81
n = 4

4, six month periods
=> 2 years

[pach2212]

Using GP this can be solved.

Every 6 months the value will become 1/3 rd of the previous value

So 81, 27, 9, 3, 1.

From the series we can see that it took 4 transitions

So 4 half years = 2 years.

[mxr55820]

Arjmen,

Could you please explain you approach. Is this a formula? If so, would you please explain the formula?

Thanks a lot.

[arjmen]

Initial Price: $81
The price falls by 66% over every 6 month period.
Now 66% ~ 2/3
So the price falls by 2/3 of the previous period's price every 6 months.

1 period = 6 months
After 1st period: New Price = 81 - (2/3)*81 = 81*[1-(2/3)]
After 2nd period: New Price = 81*[1-(2/3)]*[1-(2/3)] = 81*[1-(2/3)]^2
=> After nth period: New Price = 81*[1-(2/3)]^n

Now we know that after the nth period, Price = $1

=> 81*[1-(2/3)]^n = 1
Solve to get n = 4 periods = 24 months = 2 years

[mxr55820]

It makes sense now. Thank you for explaining