Suppose an investment of $500 doubles in value every 15 years. How much is the investment worth after 30 years? After 45 years?
please help!!! i don't even understand how to approach this problem! =(
after 30 yrs the investment is worth $2,000
after 45 yrs the investment is worth $4,000
Well, let's take a look at this investment situation. If the investment doubles in value every 15 years, we can think of it as a "doubling period."
After 15 years, the investment would be worth $1,000 since it doubles in value.
After another 15 years, so a total of 30 years, the investment would double again. This means it would be worth $2,000. (Or you can say that it doubled twice from its starting value of $500.)
Now, after 45 years, we can apply the same logic. The investment would double for the third time, making it worth $4,000. (Or you could say it doubled three times from its starting value of $500.)
So, after 30 years, the investment would be worth $2,000, and after 45 years, it would be worth $4,000. Just remember, it's all about those doubling periods!
To approach this problem, you can use the compound interest formula, which is expressed as:
A = P(1 + r/n)^(nt)
Where:
A is the final amount (investment worth)
P is the principal amount (initial investment)
r is the annual interest rate expressed as a decimal
n is the number of times that interest is compounded per year
t is the number of years
In this case, the investment doubles in value every 15 years. Therefore, the interest rate (r) would be calculated as:
r = 2^(1/15) - 1
Let's calculate the investment worth after 30 years and 45 years:
After 30 years:
P = $500
r = 2^(1/15) - 1
n = 1 (interest is compounded once per year)
t = 30 years
A = 500(1 + 2^(1/15) - 1)^(1 * 30)
After performing the calculations, the investment would be worth:
A ≈ $4,943.02
After 45 years:
P = $500
r = 2^(1/15) - 1
n = 1 (interest is compounded once per year)
t = 45 years
A = 500(1 + 2^(1/15) - 1)^(1 * 45)
After performing the calculations, the investment would be worth:
A ≈ $30,785.56
Therefore, after 30 years, the investment would be worth approximately $4,943.02, and after 45 years, it would be worth approximately $30,785.56.
To solve this problem, we need to use the concept of compound interest. Compound interest refers to the calculation of interest on both the initial investment and the accumulated interest from previous periods.
In this scenario, we are given that the investment doubles in value every 15 years. Let's break down the problem step by step:
Step 1: Determine the number of doubling periods within the given time period.
Since the investment doubles in value every 15 years, we can divide the total number of years by 15 to find out how many doubling periods have occurred.
30 years / 15 years = 2 doubling periods
45 years / 15 years = 3 doubling periods
Step 2: Calculate the investment value after each doubling period.
To find the investment value after each doubling period, we multiply the initial investment by 2 for each doubling period.
For 30 years:
Initial investment: $500
After the first 15 years: $500 * 2 = $1000
After the second 15 years: $1000 * 2 = $2000
The investment is worth $2000 after 30 years.
For 45 years:
Initial investment: $500
After the first 15 years: $500 * 2 = $1000
After the second 15 years: $1000 * 2 = $2000
After the third 15 years: $2000 * 2 = $4000
The investment is worth $4000 after 45 years.
Therefore, after 30 years, the investment is worth $2000, and after 45 years, the investment is worth $4000.
Initial investment = $500
Investment value after 15 years
= $500*2
= $1000 (doubles)
15 more years later (30), value of investment
= $1000 * 2
= $2000 (doubles from 15 years ago)
I will leave it to you to calculate the value of investment at 45 years from the initial investment.
Mathematically, let P = initial principal,
2P = PR^15 (R=interest rate)
R=e^(ln(2)/15)=1.047294123
Value of investment after N years,
V(N)=$500*R^N
for N=15,
V(15)=$500*1.047294123^15
=$1000.00
V(30)=$500*1.047294123^30
=$2000.00
and so on.