Two friends decide to invest some money. Paul invests R 1 500 every 3 months into a bank that pays 10% p.a. interest every quarter for 15 years. If Peter invests R 3 500 every year into a different bank that pays 15.5% p.a. How long will he have to invest (in years) in order to accumulate at least as much as Paul?

Answer 1

First, let's calculate how much Paul will have accumulated after 15 years of investing R 1 500 every 3 months at 10% p.a. interest every quarter.

Since the interest is compounded quarterly, the effective quarterly interest rate is 10%/4 = 2.5%.

The total number of quarters in 15 years is 15 * 4 = 60 quarters.

The formula to calculate the future value of an investment with compound interest is:

Future Value = Principal * (1 + interest rate)^n

Where:
- Principal = R 1 500
- Interest rate = 2.5%
- n = 60 quarters

Using the formula, the future value of Paul's investment after 15 years is:

FV_Paul = R 1 500 * (1 + 0.025)^60
FV_Paul = R 1 500 * (1.025)^60
FV_Paul = R 1 500 * 3.348
FV_Paul = R 5 022

Now, let's calculate how much Peter will have accumulated if he invests R 3 500 every year at 15.5% p.a. interest, and he invests for x number of years.

The future value formula here is similar:

FV_Peter = R 3 500 * (1 + 0.155)^x

Now we need to find out for how many years Peter needs to invest in order to accumulate at least as much as Paul, i.e., FV_Peter ≥ FV_Paul.

Let's substitute the values we know:

R 3 500 * (1.155)^x ≥ R 5 022

Solving for x:

(1.155)^x ≥ 5 022 / 3 500
(1.155)^x ≥ 1.4357

Take the natural log of both sides to solve for x:

x * ln(1.155) ≥ ln(1.4357)
x * 0.1472 ≥ 0.362

Now solve for x:

x ≥ 0.362 / 0.1472
x ≥ 2.459

Peter needs to invest for at least 3 years to accumulate as much as Paul.