Nicole is back at it again and wants to see, this time, if listening to classical music while taking a test affects test scores.
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Compound Interest
A total of $20,000 is invested in different assets.
45% is invested in a Treasury bond for 3 years at 4.35 APR compounded annually.
For this investment, the principal is P = 0.45*$20,000 = $9,000.
The compounding period is yearly, thus the interest rate is:
i = 4.35 / 100 = 0.0435
The duration (in periods) is n = 3
Calculate the final value with the formula:
Substituting:
The second investment is a CD at 3.75% APR for 3 years compounded annually. The parameters for the calculations are as follows:
P = 15% of $20,000 = $3,000
i = 3.75 / 100 = 0.0375
n = 3
Calculating:
The third investment is in a stock plan. The initial value of the investment is
P = 20% of $20,000 = $4,000
By the end of the first year, the stock plan increased by 8%, thus its value is:
M1 = $4000 * 1.2 = $4,800
By the end of the second year, the stock plan decreased by 4$, thus the value is:
M2 = $4,800 * 0.96 = $4,608
Finally, the stock plan increases by 6%, resulting in a final balance of:
M3 = $4,608 * 1.06 = $4,884.48
Finally, the last investment is in a savings account at 2.90% APR compounded annually for 3 years (not mentioned, but assumed).
P = $20,000 - $9,000- $3,000 - $4,000 = $4,000
i = 2.90 / 100 = 0.029
n = 3
Calculating:
To summarize, the final balances for each type of investment at the end of the third year are:
Investment 1; $10,226.33
Investment 2: $3,350.31
Investment 3: $4,884.48
Investment 4: $4,358.19
Total balance: $22,819.32
We'll use the following properties of sine and cosine to prove this:
Then it's just a matter of filling it in...
sin20sin40 * sin60sin80 = 1/2(cos20 - cos60) * 1/2 (cos20 - cos140) =
1/8( cos40 + 1 - cos160 - cos120 - cos40 - cos80 + cos80 + cos200) =
1/8(1 - cos160 - cos120 + cos200) =
1/8(1 - cos160 - cos120 + cos160) =
1/8(1 - cos120 ) = 1/8( 1 + 1/2 ) = 3/16
Then it's just a matter of filling it in...
sin20sin40 * sin60sin80 = 1/2(cos20 - cos60) * 1/2 (cos20 - cos140) =
1/8( cos40 + 1 - cos160 - cos120 - cos40 - cos80 + cos80 + cos200) =
1/8(1 - cos160 - cos120 + cos200) =
1/8(1 - cos160 - cos120 + cos160) =
1/8(1 - cos120 ) = 1/8( 1 + 1/2 ) = 3/16