Definition: Simple Interest
Simple interest is where you earn interest on the initial amount that you invested, but not interest on interest.
As an easy example of simple interest, consider how much you will get by investing R1 000 for 1 year with a bank that pays you 5% simple interest. At the end of the year, you will get an interest of: Interest = R1 000×5% = R1 000× 5 100 = R1 000×0,05 = R50
So, with an “opening balance” of R1 000 at the start of the year, your “closing balance” at the end of the year will therefore be:
Closing Balance = Opening Balance+Interest
= R1 000 + R50 = R1 050
We sometimes call the opening balance in financial calculationsPrincipal, which is abbreviated asP (R1 000 in the example). The interest rate is usually labelledi(5% in the example), and the interest amount (in Rand terms) is labelledI (R50 in the example).
So we can see that:
I=P×i (8.1)
and
Closing Balance = Opening Balance+Interest
= P+I
= P+ (P×i) = P(1 +i)
This is how you calculate simple interest. It is not a complicated formula, which is just as well because you are going to see a lot of it!
Not Just One
You might be wondering to yourself:
1. how much interest will you be paid if you only leave the money in the account for 3 months, or
2. what if you leave it there for 3 years?
It is actually quite simple - which is why they call itSimple Interest.
1. Three months is 1/4 of a year, so you would only get 1/4 of a full year’s interest, which is: 1/4×(P×i). The closing balance would therefore be:
Closing Balance = P + 1/4×(P×i) = P(1 + (1/4)i)
2. For 3 years, you would get three years’ worth of interest, being: 3×(P ×i). The closing balance at the end of the three year period would be:
Closing Balance = P+ 3×(P×i) = P×(1 + (3)i)
If you look carefully at the similarities between the two answers above, we can generalise the result. In other words, if you invest your money (P) in an account which pays a rate of interest (i) for a period of time (nyears), then, using the symbol (A) for the Closing Balance:
Closing Balance,(A)=P(1 +i·n) (8.2) As we have seen, this works whennis a fraction of a year and also whenncovers several years.
Important: Interest Calculation
Annual Rates means Yearly rates. and p.a.(per annum) = per year
Worked Example 9: Simple Interest
Question: If I deposit R1 000 into a special bank account which pays a Simple Interest of 7% for 3 years, how much will I get back at the end?
Answer
Step 1 : Determine what is given and what is required
• opening balance,P = R1 000
• interest rate,i= 7%
• period of time,n= 3 years
We are required to find the closing balance (A). Step 2 : Determine how to approach the problem We know from (8.2) that:
Step 3 : Solve the problem
A = P(1 +i·n)
= R1 000(1 + 3×7%) = R1 210
Step 4 : Write the final answer
The closing balance after 3 years of saving R1 000 at an interest rate of 7% is R1 210.
Worked Example 10: Calculating n
Question: If I deposit R30 000 into a special bank account which pays a Simple Interest of 7.5% ,for how many years must I invest this amount to generate R45 000 Answer
Step 1 : Determine what is given and what is required
• opening balance,P = R30 000
• interest rate,i= 7,5%
• closing balance,A= R45 000
We are required to find the number of years.
Step 2 : Determine how to approach the problem We know from (8.2) that:
Closing Balance (A)=P(1 +i·n)
Step 3 : Solve the problem
Closing Balance (A) = P(1 +i·n)
R45 000 = R30 000(1 +n×7,5%) (1 + 0,075×n) = 45000 30000 0,075×n = 1,5−1 n = 0,5 0,075 n = 6,6666667
Step 4 : Write the final answer
nhas to be a whole number, thereforen= 7.
The period is 7 years for R30 000 to generate R45 000 at a simple interest rate of 7,5%.
8.4.1
Other Applications of the Simple Interest Formula
Worked Example 11: Hire-Purchase
Question: Troy is keen to buy an addisional hard drive for his laptop advertised for R 2 500 on the internet. There is an option of paying a 10% deposit then making 24 monthly payments using a hire-purchase agreement where interest is calculated at 7,5% p.a. simple interest. Calculate what Troy’s monthly payments will be. Answer
Step 1 : Determine what is given and what is required
A new opening balance is required, as the 10% deposit is paid in cash.
• 10% of R 2 500 = R250
• new opening balance,P = R2 500−R250 = R2 250
• interest rate,i= 7,5% = 0,075pa
• period of time,n= 2 years
We are required to find the closing balance (A) and then the montly payments. Step 2 : Determine how to approach the problem
We know from (8.2) that:
Closing Balance,(A)=P(1 +i·n)
Step 3 : Solve the problem
A = P(1 +i·n)
= R2 250(1 + 2×7,5%) = R2 587,50
Monthly payment = 2587,50÷24 = R107,81
Step 4 : Write the final answer Troy’s monthly payments = R 107,81
Worked Example 12: Depreciation
Question: Seven years ago, Tjad’s drum kit cost him R12 500. It has now been valued at R2 300. What rate of simple depreciation does this represent ?
Answer
Step 1 : Determine what is given and what is required
• opening balance,P = R12 500
• period of time,n= 7 years
• closing balance,A= R2 300
We are required to find the rate(i).
Step 2 : Determine how to approach the problem We know from (8.2) that:
Closing Balance,(A)=P(1 +i·n)
Therefore, fordepreciationthe formula will change to: Closing Balance,(A)=P(1−i·n)
Step 3 : Solve the problem
A = P(1−i·n)
R2 300 = R12 500(1−7×i)
i = 0,11657...
Step 4 : Write the final answer
Therefore the rate of depreciation is11,66%
Exercise: Simple Interest
1. An amount of R3 500 is invested in a savings account which pays simple interest at a rate of 7,5% per annum. Calculate the balance accumulated by the end of 2 years.
2. Calculate the simple interest for the following problems. (a) A loan of R300 at a rate of 8% for l year.
(b) An investment of R225 at a rate of 12,5% for 6 years.
3. I made a deposit of R5 000 in the bank for my 5 year old son’s 21st birthday. I have given him the amount of R 18 000 on his birtday. At what rate was the money invested, if simple interest was calculated ?
4. Bongani buys a dining room table costing R 8 500 on Hire Purchase. He is charged simple interest at 17,5% per annum over 3 years.
(a) How much will Bongani pay in total ? (b) How much interest does he pay ? (c) What is his montly installment ?