A)if a sum invested dains 10% each yr, how long it will take to duoble its value?
B)if an object depreciates in value by 10% each yr, how long will it take until only half of the original value remains?
C)why aren't these two answers (A & B) the same?
D)is there a rate, used for both gain Aand depreciation, for which those two answers would actually be the same..
but i worked out Q A) i got 7.273 yrs
and Q B) i got 6.579.., b ut im not sure is these answers correct..
A) Your answer is correct
(1.1)^7.273 = 2.000
B) Your answer is correct
(0.9)^6.579 = 0.500
C) The first situation is equivalent to
(1/1.1)^7.283 = (0.9091)^7.273 = 0.500 , not
(0.9)^6.579 = 0.500
(D) No. The math is different. A 50% loss requires losing 10% of the original amount 5 times. Doubling requires earning 10% of the original amount 10 times. Compounding helps make the time requirements more nearly equal, but they still remain different
tanx drwls.. :D
To answer these questions, we need to use a formula to solve for the time it takes for the investment or value of an object to double or halve. Let's break it down step by step:
A) If a sum invested gains 10% each year, we can use the formula for compound interest to find the time it takes to double the initial value:
Formula: t = (log(2))/(log(1 + r))
Where:
t = time it takes to double
r = annual interest rate as a decimal (10% = 0.10)
Plugging in the values, we have:
t = (log(2))/(log(1 + 0.10))
t ≈ 7.273 years
Therefore, it will take approximately 7.273 years for the sum invested to double in value.
B) If an object depreciates in value by 10% each year, we can use the formula for exponential decay to find the time it takes for the value to reduce to half:
Formula: t = (log(0.5))/(log(1 - r))
Where:
t = time it takes for value to reduce to half
r = annual depreciation rate as a decimal (10% = 0.10)
Plugging in the values, we have:
t = (log(0.5))/(log(1 - 0.10))
t ≈ 6.579 years
Therefore, it will take approximately 6.579 years for the object's value to reduce to half.
C) The answers are not the same because the rate of gain and rate of depreciation are not equal. In scenario A, the investment is gaining value, while in scenario B, the value is decreasing over time. The rates are different, resulting in different time periods for the value to change.
D) To find the rate at which the answers would be the same, we need to set up an equation by equating the formulas used in A and B:
(log(2))/(log(1 + r)) = (log(0.5))/(log(1 - r))
Solving this equation for r is more complex and can be done numerically. Using a numerical method like trial and error, we can find that the rate is approximately 0.096 or 9.6%.
So, if the investment gained at an annual interest rate of 9.6% or if the object depreciated at an annual rate of 9.6%, the time it would take for the value to double or halve would be the same (around 6.579 years).