Tom and Lana both need $15,000 in 3 years. Tom finds an investment in which he can earn 8% compounded annually. Lana finds an investment in which she earns 7 7/8% interest compounded monthly. Who must invest more money now in order to get $15,000 in 3 years? How much more?
___(Tom/Lana)___ has to invest $________ more than the other one in order to have the same amount ($15,000) in 3 years.
Tom:
Paym(1.08)^3 = 15000
Paym = 15000/1.08^3 = $11,907.48
Lana: monthly rate = .07875/12 = .0065625
Paym(1.0065625)^36 = 15000
Paym = 15000/1.0065625^36 = $11,852.89
etc
To determine who must invest more money, we need to compare the future value (amount at the end of 3 years) of their investments.
For Tom:
Principal (P) = ?
Interest Rate (R) = 8%
Time (t) = 3 years
Future Value (FV) = $15,000
Using the formula for compound interest:
FV = P(1 + (R/100))^t
Plugging in the values:
$15,000 = P(1 + (8/100))^3
Simplifying the equation:
(1.08)^3 = P
1.2597 = P
So, Tom must invest $1,2597.
For Lana:
Principal (P) = ?
Interest Rate (R) = 7 7/8% = 7.875%
Time (t) = 3 years
Future Value (FV) = $15,000
Since Lana's interest is compounded monthly, we need to adjust the interest rate and time period accordingly:
Monthly Interest Rate = 7.875% / 12 = 0.65625%
Months = 12 * 3 = 36 months
Using the same formula:
FV = P(1 + (R/100))^t
Plugging in the adjusted values:
$15,000 = P(1 + (0.65625/100))^36
Simplifying the equation:
(1.0065625)^36 = P
1.28163 = P
So, Lana must invest $1.28163.
To determine who must invest more, we compare the amounts:
Tom has to invest $1,2597, and Lana has to invest $1.28163.
Comparing the investments:
Tom's investment: $1,2597
Lana's investment: $1.28163
Therefore, Tom must invest $1,2597 - $1.28163 = $-0.02106 more than Lana.
However, since negative money doesn't make sense, we can conclude that both Tom and Lana need to invest the same amount of money to have $15,000 in 3 years.
To determine who must invest more money, we need to compare the future values of their investments after 3 years. The future value of an investment can be calculated using the formula:
Future Value = Principal (1 + Interest Rate)^Time
For Tom:
Principal (P1) = ?
Interest Rate (r1) = 8% = 0.08
Time (t) = 3 years
Future Value (F1) = $15,000
Using the given formula, we can rearrange it to solve for P1:
P1 = F1 / (1 + r1)^t
Plug in the values:
P1 = $15,000 / (1 + 0.08)^3
P1 = $15,000 / 1.259712
P1 ≈ $11,914.77
For Lana:
Principal (P2) = ?
Interest Rate (r2) = 7 7/8% = 7.875% = 0.07875
Time (t) = 3 years
Future Value (F2) = $15,000
Using the same formula as before, we rearrange it to solve for P2:
P2 = F2 / (1 + r2)^t
Plug in the values:
P2 = $15,000 / (1 + 0.07875)^3
P2 = $15,000 / 1.254789
P2 ≈ $11,950.35
Therefore, Lana has to invest more money than Tom in order to have the same amount ($15,000) in 3 years. The difference in their investments can be calculated by subtracting Tom's investment (P1) from Lana's investment (P2):
Difference = P2 - P1
Difference = $11,950.35 - $11,914.77
Difference ≈ $35.59
Lana must invest approximately $35.59 more than Tom in order to have the same amount ($15,000) in 3 years.