(COMPOUND INTEREST)

Effie is planning to set up a trust fund for her new grand daughter. In 18 years time she wants the investment to be worth $50000 to help her buy a car or a home unit.
a)How much should she invest at 5% p.a. compounding yearly?
--I know the answer for this is $20776.03

b) How much less will she have to invest at 6% p.a., compounding monthly?
This is the question I need help with...

Does anyone know the working out and answer? thank you

let her present investment be P

P(1.05)^18 = 50000
P = 50000/(1.05)^18 = 20776.03

for the 2nd part,
monthly rate = .06/12 = .005, but the exponent = 12x18 = 216

P = 50000/1.005^216 = 17025.53

To find out how much Effie should invest at 5% p.a. compounding yearly, we can use the compound interest formula:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment
P = the principal amount (the initial investment)
r = the annual interest rate (as a decimal)
n = the number of compounding periods per year
t = the number of years

Given that Effie wants the investment to be worth $50,000 in 18 years, we can plug in the values into the formula:

50000 = P(1 + 0.05/1)^(1*18)

To solve for P, we divide both sides by the bracket term:

P = 50000 / (1.05)^18

P ≈ $20,776.03

Now, moving on to the second question.

b) To find out how much less she will have to invest at 6% p.a. compounding monthly, we follow the same approach. However, since the compounding is happening monthly, the values for r and n will be different.

In this case, r = 0.06 (6% expressed as a decimal) and n = 12 (since compounding is happening monthly).

So, the new formula would be:

50000 = P(1 + 0.06/12)^(12*18)

Simplifying,

P = 50000 / (1.005)^216

P ≈ $17,152.15

Therefore, Effie will have to invest approximately $17,152.15 less at 6% p.a. compounding monthly compared to 5% p.a. compounding yearly for the investment to be worth $50,000 in 18 years.