# 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

\$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.