Calculations Using Simple and Compound Interest Revision Notes for Grade 10 NSC Matric Mathematics

Calculations Using Simple and Compound Interest

Introduction to financial calculations

Financial calculations help you make informed decisions about money in real-life situations. You need to understand two main types of interest calculations: simple interest and compound interest. Each type is used in different financial scenarios.

Hire purchase

What is hire purchase?

Hire purchase is a financial agreement that allows you to buy items without paying the full cash price immediately. Instead of borrowing money from a bank, you make an arrangement directly with the shop to pay in instalments.

Key features of hire purchase:

You usually pay a deposit (percentage of the cash price)
The remaining amount becomes your loan
Interest is charged on this loan amount
You make equal monthly payments over an agreed period

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Critical exam point: Hire purchase agreements always use simple interest, not compound interest. This is because the agreement has fixed terms from the beginning.

How to calculate hire purchase payments

Follow these steps for any hire purchase calculation:

Find the principal amount: Principal = Cash price - Deposit
Apply the simple interest formula: $A = P(1 + in)$
Calculate monthly payments: Monthly payment = Total amount ÷ Number of months

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Key Variables:

$P$ = Principal (loan amount after deposit)
$i$ = Interest rate (as a decimal)
$n$ = Time period (in years)
$A$ = Total amount to be repaid

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Worked Example: Basic hire purchase calculation

Question: Troy wants to purchase a computer screen advertised for R 2500. He can pay a 10% deposit and make 24 monthly payments at 7,5% per annum simple interest. Calculate his monthly payment.

Solution:

Step 1: Calculate the deposit and principal

Deposit = 10% of R 2500 = R 250
Principal amount = R 2500 - R 250 = R 2250

Step 2: Identify the variables

$P$ = R 2250
$i$ = 7,5% = 0,075
$n$ = 24 months = 2 years

Step 3: Use simple interest formula

$A = P(1 + in)$
$A = 2250(1 + 0,075 × 2)$
$A = 2250 × 1,15 = R 2587,50$

Step 4: Find monthly payment

Monthly payment = R 2587,50 ÷ 24 = R 107,81

Answer: Troy's monthly payment will be R 107,81.

Hire purchase with additional costs

Sometimes hire purchase agreements include insurance premiums added to each monthly payment. This insurance protects both you and the seller if payments are missed.

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Worked Example: Hire purchase with insurance

Question: Cassidy purchases a TV with a cash price of R 5500. She pays it off over 54 months at an interest rate of 21% per annum. An insurance premium of R 12,50 is added to every monthly payment. Calculate her total monthly payment.

Solution:

Step 1: Identify the variables (no deposit mentioned)

$P$ = R 5500
$i$ = 21% = 0,21
$n$ = 54 months = 4,5 years

Step 2: Apply simple interest formula

$A = P(1 + in)$
$A = 5500(1 + 0,21 × 4,5)$
$A = 5500 × 1,945 = R 10 697,50$

Step 3: Calculate basic monthly payment

Monthly payment = R 10 697,50 ÷ 54 = R 198,10

Step 4: Add insurance premium

Total monthly payment = R 198,10 + R 12,50 = R 210,60

Answer: Cassidy will pay R 210,60 per month for 54 months.

Inflation calculations

Understanding inflation

Inflation represents the average increase in prices of goods and services each year, expressed as a percentage. Unlike hire purchase, inflation calculations use compound interest because price increases build upon previous increases year after year.

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Why compound interest for inflation? Inflation uses compound interest because each year's price increase is calculated on the already-increased price from the previous year, creating a compounding effect.

Calculating future costs with inflation

To determine what something will cost in the future due to inflation, use the compound interest formula:

Formula: $A = P(1 + i)^n$

Where:

$P$ = Current price
$i$ = Inflation rate (as a decimal)
$n$ = Number of years
$A$ = Future price

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Worked Example: Future cost calculation

Question: Milk currently costs R 14 for two litres. How much will it cost in 4 years if the inflation rate is 9% per annum?

Solution:

Step 1: Write down known variables

$P$ = R 14
$i$ = 9% = 0,09
$n$ = 4 years

Step 2: Apply compound interest formula

$A = P(1 + i)^n$
$A = 14(1 + 0,09)^4$
$A = 14 × (1,09)^4 = R 19,76$

Answer: In four years, two litres of milk will cost R 19,76.

Calculating past costs with inflation

To find what something cost in the past, rearrange the compound interest formula to solve for $P$ :

Rearranged formula: $P = \frac{A}{(1 + i)^n}$

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Worked Example: Past cost calculation

Question: A box of chocolates costs R 55 today. How much did it cost 3 years ago if the average inflation rate was 11% per annum?

Solution:

Step 1: Write down known variables

$A$ = R 55 (current price)
$i$ = 11% = 0,11
$n$ = 3 years

Step 2: Rearrange and solve for past price

$55 = P(1 + 0,11)^3$
$P = 55 ÷ (1,11)^3$
$P = 55 ÷ 1,367 = R 40,22$

Answer: Three years ago, the box of chocolates cost R 40,22.

Population growth calculations

Why population growth uses compound interest

Population growth follows an exponential pattern because each person born has the potential to start a family in the future. This creates growth that compounds over time, making the compound interest formula appropriate.

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Population Growth Pattern Unlike hire purchase (which has fixed terms), population growth is exponential because:

New births each year add to the existing population
These new people can also reproduce in future years
This creates a compounding effect similar to compound interest

Population growth formula

Use the same compound interest formula as for inflation:

Formula: $A = P(1 + i)^n$

Where:

$P$ = Current population
$i$ = Growth rate (as a decimal)
$n$ = Number of years
$A$ = Future population

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Worked Example: Population growth

Question: The current population of Johannesburg is 3 888 180, and the average population growth rate in South Africa is 2,1% per annum. What can city planners expect the population of Johannesburg to be in 10 years?

Solution:

Step 1: Write down known variables

$P$ = 3 888 180
$i$ = 2,1% = 0,021
$n$ = 10 years

Step 2: Apply compound interest formula

$A = P(1 + i)^n$
$A = 3 888 180(1 + 0,021)^{10}$
$A = 3 888 180 × (1,021)^{10} = 4 786 343$

Answer: City planners can expect Johannesburg's population to be 4 786 343 in ten years.

When to use simple vs compound interest

Application	Interest Type	Reason
Hire purchase	Simple interest	Fixed loan agreement terms
Inflation	Compound interest	Prices build on previous increases
Population growth	Compound interest	Exponential growth pattern
Bank savings	Compound interest	Earn interest on accumulated interest

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Memory Aid: Remember "Hire = Simple" - hire purchase agreements always use simple interest because the terms are fixed from the start. Everything that grows exponentially (inflation, population, savings) uses compound interest.

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Key Points to Remember:

Hire purchase always uses simple interest - this is the most important exam rule to remember
Inflation and population growth use compound interest because they grow exponentially over time
Simple interest formula: $A = P(1 + in)$ where $n$ is time in years
Compound interest formula: $A = P(1 + i)^n$ where the exponent shows compounding effect
Monthly payment calculation: Always divide the total accumulated amount by the number of months

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Exam Success Tips:

Always identify whether the question requires simple or compound interest first
Convert all percentages to decimals by dividing by 100
Ensure time periods are in the correct units (years for the formulas shown)
For hire purchase, remember to subtract any deposit from the cash price
Follow rounding instructions carefully
Add insurance premiums after calculating basic monthly payments

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Common Exam Traps to Avoid:

Forgetting that hire purchase uses simple interest
Not converting months to years when using annual interest rates
Forgetting to subtract deposits in hire purchase calculations
Mixing up future and past cost calculations for inflation

Calculations Using Simple and Compound Interest (Grade 10 NSC Matric Mathematics): Revision Notes