Finance, Growth, and Decay Revision Notes for Grade 11 NSC Matric Mathematics

Simple and Compound Depreciation

What is depreciation?

Depreciation is the decrease in value of an asset over time due to usage, age, and wear. When you buy a new car, it immediately loses value when you drive it off the lot - this is depreciation in action.

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Common assets that depreciate include:

Vehicles (cars, trucks, tractors)
Equipment (computers, machinery, furniture)
Buildings and office equipment

Businesses use depreciation calculations for:

Tax purposes - depreciation reduces taxable income
Asset valuation - determining current worth of company assets
Financial planning - budgeting for asset replacement

Types of depreciation

There are two main methods for calculating depreciation:

Simple depreciation (also called straight-line depreciation)
Compound depreciation (also called reducing-balance depreciation)

Simple depreciation (straight-line method)

Definition and explanation

In simple depreciation, an asset loses the same rand amount of value each year. The depreciation amount is calculated once at the beginning and remains constant throughout the asset's life.

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Key characteristics:

Constant rand amount lost each year
Based on the original purchase price
Creates a straight line when graphed
Simpler to calculate and understand

Simple depreciation formula

$A = P(1 - in)$

Where:

A = final book value (depreciated value)
P = principal amount (original value)
i = depreciation rate (as a decimal)
n = time period in years

Alternative formula for total depreciation: $\text{Total depreciation} = P \times i \times n$

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Worked Example: Simple depreciation calculation

Question: A smartphone costs R 6000 and depreciates at 22% per annum on a straight-line basis. Find the value after 4 years.

Solution:

Step 1: Calculate the annual depreciation amount

$\text{Depreciation} = 6000 \times \frac{22}{100} = \text{R } 1320 \text{ per year}$

Step 2: Create a depreciation schedule

Year	Value at beginning	Depreciation	Value at end
1	R 6000	R 1320	R 4680
2	R 4680	R 1320	R 3360
3	R 3360	R 1320	R 2040
4	R 2040	R 1320	R 720

Step 3: Use the formula to verify

$A = P(1 - in) = 6000(1 - 0,22 \times 4) = 6000(1 - 0,88) = 6000(0,12) = \text{R } 720$

Answer: After 4 years, the smartphone is worth R 720.

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Worked Example: Another simple depreciation example

Question: A car valued at R 240 000 depreciates at 15% per annum using straight-line depreciation. Calculate the value after 5 years.

Solution:

Step 1: Write down known variables and formula

P = 240 000
i = 0,15
n = 5
A = P(1 - in)

Step 2: Substitute values

$A = 240 000(1 - 0,15 \times 5) = 240 000(1 - 0,75) = 240 000(0,25) = 60 000$

Answer: After 5 years, the car is worth R 60 000.

Compound depreciation (reducing-balance method)

Definition and explanation

In compound depreciation, an asset loses the same percentage of its current value each year. The depreciation amount decreases each year because it's calculated on the reducing book value.

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Key characteristics:

Constant percentage lost each year
Based on the current (reducing) value
Depreciation amount gets smaller each year
Asset never reaches zero value
More realistic for most assets

Compound depreciation formula

$A = P(1 - i)^n$

Where:

A = final book value (depreciated value)
P = principal amount (original value)
i = depreciation rate (as a decimal)
n = time period in years

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Note: This formula is similar to compound interest: $A = P(1 + i)^n$ , but with subtraction instead of addition.

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Worked Example: Compound depreciation calculation

Question: A farm tractor worth R 60 000 depreciates at 20% per annum on a reducing-balance basis. Find the value after 5 years.

Solution:

Step 1: Write down known variables

P = 60 000
i = 0,2
n = 5

Step 2: Create a depreciation schedule

Year	Book value	Depreciation	Value at end
1	R 60 000	60 000 × 0,2 = 12 000	R 48 000
2	R 48 000	48 000 × 0,2 = 9600	R 38 400
3	R 38 400	38 400 × 0,2 = 7680	R 30 720
4	R 30 720	30 720 × 0,2 = 6144	R 24 576
5	R 24 576	24 576 × 0,2 = 4915,20	R 19 660,80

Step 3: Use the formula to verify

$A = P(1 - i)^n = 60 000(1 - 0,2)^5 = 60 000(0,8)^5 = 60 000(0,32768) = \text{R } 19 660,80$

Answer: After 5 years, the tractor is worth R 19 660,80.

Key observations about compound depreciation

When calculating reducing-balance depreciation:

The depreciation amount changes each year
The depreciation amount gets smaller each year
The book value at year-end becomes the principal for the next year
The asset always retains some value (never reaches zero)

Comparing simple and compound depreciation

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Worked Example: Method comparison

Question: A school buys a minibus for R 950 000 that depreciates at 13,5% per annum. Compare the values after 3 years using both methods.

Simple depreciation:

$A = 950 000(1 - 3 \times 0,135) = 950 000(0,595) = \text{R } 565 250$

Compound depreciation:

$A = 950 000(1 - 0,135)^3 = 950 000(0,865)^3 = \text{R } 614 853,89$

Conclusion: After 3 years, simple depreciation results in a lower value (R 565 250) than compound depreciation (R 614 853,89).

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This is because simple depreciation calculates on the full original value each year, while compound depreciation calculates on the reducing balance. The graph shows how asset values decline differently under each method, with simple depreciation creating a straight line and compound depreciation creating a curve.

Finding the depreciation rate

Sometimes you need to work backwards to find the depreciation rate when you know the original and final values.

Finding the rate for simple depreciation

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Worked Example: Finding simple depreciation rate

Question: After 4 years, a computer's value is halved. Assuming simple decay, find the annual depreciation rate.

Solution:

Step 1: Set up the equation

Let the original value be x, so:

$A = \frac{x}{2}, P = x, n = 4$

$\frac{x}{2} = x(1 - 4i)$

Step 2: Solve for i

$\frac{1}{2} = 1 - 4i$
$4i = 1 - \frac{1}{2} = \frac{1}{2}$
$i = \frac{1}{8} = 0,125$

Answer: The computer depreciated at 12,5% per annum.

Finding the rate for compound depreciation

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Worked Example: Finding compound depreciation rate

Question: Cristina bought a fridge for R 8999 and sold it 3 years later for R 4500. Find the annual depreciation rate assuming reducing-balance method.

Solution:

Step 1: Write the compound decay formula

$A = P(1 - i)^n$
$4500 = 8999(1 - i)^3$

Step 2: Solve for i

$\frac{4500}{8999} = (1 - i)^3$
$\sqrt[3]{\frac{4500}{8999}} = 1 - i$
$i = 1 - \sqrt[3]{\frac{4500}{8999}} = 1 - \sqrt[3]{0,5} = 1 - 0,794 = 0,206$

Answer: The fridge depreciated at 20,6% per annum.

Exam tips and common mistakes

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Calculator usage

Always do calculations in one step on your calculator
Don't round intermediate answers - only round the final answer
Round monetary amounts to two decimal places (cents)

Common exam traps

Don't confuse simple and compound depreciation formulas
Check whether the rate is given as a percentage or decimal
Remember that in compound depreciation, the asset never reaches zero value
Be careful with the time period - ensure you're using the correct number of years

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Method comparison

Simple depreciation usually gives a lower final value
Compound depreciation is more realistic for most assets
The difference increases over longer time periods

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Key Points to Remember:

Simple depreciation: Same rand amount lost each year - use $A = P(1 - in)$
Compound depreciation: Same percentage lost each year - use $A = P(1 - i)^n$
Simple creates a straight line graph, compound creates a curve
Always verify your answer makes sense - depreciated value should be less than original value
Practice both methods as exam questions often ask you to compare them

Finance, Growth, and Decay (Grade 11 NSC Matric Mathematics): Revision Notes

Simple and Compound Depreciation

What is depreciation?

Types of depreciation

Simple depreciation (straight-line method)

Definition and explanation

Simple depreciation formula

Compound depreciation (reducing-balance method)

Definition and explanation

Compound depreciation formula

Key observations about compound depreciation

Comparing simple and compound depreciation

Finding the depreciation rate

Finding the rate for simple depreciation

Finding the rate for compound depreciation

Exam tips and common mistakes

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