An investment of $4000 is deposited into an account in which interest is compounded monthly. Complete the table by filling in the amounts to which the investment grows at the indicated times. (Round your answers to the nearest cent.)
r = 4%
$200
Well, when it comes to investments, it's no laughing matter! But let me help you fill in the table with a touch of humor. Here we go:
Time (years) Amount (dollars)
0 $4000 (Original investment - no growth, just like my hairline)
1 $4160.32 (Hey, it's growing, just like my stack of dad jokes!)
2 $4323.58 (Looks like your investment is multiplying faster than rabbits!)
3 $4490.99 (Wow, your money is growing like a weed...but a good kind of weed!)
4 $4662.79 (The growth of your investment is starting to look like my collection of clown noses!)
Remember to round to the nearest cent! And always keep an eye on your investment, just like I keep an eye out for slipping on banana peels on the job.
To calculate the amounts to which the investment grows at the indicated times, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest
P = the principal investment amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested for
In this case, the principal investment amount (P) is $4000, the annual interest rate (r) is 4% (or 0.04 as a decimal), and interest is compounded monthly, so n = 12.
Let's calculate the amounts for different times:
At 1 year:
A = 4000(1 + 0.04/12)^(12*1)
A = 4000(1.003333)^12
A ≈ $4,051.48
At 3 years:
A = 4000(1 + 0.04/12)^(12*3)
A = 4000(1.003333)^36
A ≈ $4,244.13
At 5 years:
A = 4000(1 + 0.04/12)^(12*5)
A = 4000(1.003333)^60
A ≈ $4,436.61
At 10 years:
A = 4000(1 + 0.04/12)^(12*10)
A = 4000(1.003333)^120
A ≈ $5,046.06
The completed table would look like this:
Time (years) | Amount ($)
-------------------------
1 | $4,051.48
3 | $4,244.13
5 | $4,436.61
10 | $5,046.06
To find the amounts to which the investment grows at the indicated times, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, the principal amount is $4000, the annual interest rate is 4% (or 0.04 as a decimal), and interest is compounded monthly (so n would be 12).
Let's complete the table:
Time | Amount (A)
----------------------------
1 year |
3 years |
5 years |
10 years |
20 years |
30 years |
To find the amount after 1 year, we substitute the values into the formula:
A = 4000(1 + 0.04/12)^(12*1)
Solving this equation, we get:
A = 4000(1.00333)^12
A ≈ 4000(1.04050699)
A ≈ $4161.43 (rounded to the nearest cent)
We repeat this process for the other time intervals, substituting the respective values into the formula:
3 years:
A = 4000(1 + 0.04/12)^(12*3)
5 years:
A = 4000(1 + 0.04/12)^(12*5)
10 years:
A = 4000(1 + 0.04/12)^(12*10)
20 years:
A = 4000(1 + 0.04/12)^(12*20)
30 years:
A = 4000(1 + 0.04/12)^(12*30)
Calculating these equations will give you the amounts to which the investment grows at the indicated times, rounded to the nearest cent.
4000(1+interest rate/12 months)^12months
4000(1+0.04/12)^12 = 4162.966172
rounding = 4162.97