Declining-Balance Depreciation Revision Notes for HSC SSCE Mathematics Standard

Declining-Balance Depreciation

What is declining-balance depreciation?

Declining-balance depreciation is a method where an asset loses value by the same percentage in each time period. This means the actual dollar amount of depreciation gets smaller over time, even though the percentage stays constant.

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Understanding the concept with a real example:

Imagine you purchase a car for $20,000 that depreciates by 10% each year. After the first year, the car loses $2,000 in value (10% of $20,000), bringing its value down to $18,000. In the second year, it loses $1,800 (10% of $18,000), making the value $16,200. Notice how the dollar amount of depreciation decreased from $2,000 to $1,800, even though the percentage remained at 10%.

This depreciation method works similarly to compound interest calculations, except the value decreases rather than increases. Just as compound interest builds on previous interest, declining-balance depreciation compounds the loss in value.

The declining-balance depreciation formula

The formula for calculating declining-balance depreciation is:

$S = V_0(1 - r)^n$

Where each variable represents:

$S$ = Salvage value or current value of an item, also referred to as the book value
$V_0$ = Purchase price of the item when you first bought it (value when $n = 0$ )
$r$ = Rate of depreciation per time period expressed as a decimal
$n$ = Number of time periods that have passed

chatImportant

Converting percentages to decimals:

Always convert percentage rates to decimals by dividing by 100. For example, 25% becomes 0.25. This is a critical step that students often forget!

Worked examples

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Worked Example 1: Calculating the salvage value

Eva purchased a new car two years ago for $32,000. During the first year it depreciated by 25% and during the second year it depreciated by 20% of its value after the first year. What is the current value of the car?

Solution:

Step 1: Write the declining-balance depreciation formula.

$S = V_0(1 - r)^n$

Step 2: For the first year, substitute $V_0 = 32000$ , $r = 0.25$ and $n = 1$ into the formula.

$= 32000 \times (1 - 0.25)^1$

Step 3: Evaluate the value of the car after the first year.

$= \$24000$

Step 4: Write the declining-balance depreciation formula again for the second year.

$S = V_0(1 - r)^n$

Step 5: For the second year, substitute $V_0 = 24000$ , $r = 0.20$ and $n = 1$ into the formula.

$= 24000 \times (1 - 0.20)^1$

Step 6: Evaluate the value of the car after the second year.

$= \$19200$

Step 7: Write the answer in words.

Current value is $19,200.

Exam tip: When depreciation rates differ each year, calculate the value at the end of each period separately, using the previous salvage value as the new purchase price.

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Worked Example 2: Calculating the purchase price

Angus buys a car that depreciates at the rate of 26% per annum. After five years the car has a salvage value of $17,420. How much did Angus pay for the car, to the nearest dollar?

Solution:

Step 1: Write the declining-balance depreciation formula.

$S = V_0(1 - r)^n$

Step 2: Substitute $S = 17420$ , $r = 0.26$ and $n = 5$ into the formula.

$17420 = V_0 \times (1 - 0.26)^5$

Step 3: Make $V_0$ the subject of the equation by dividing both sides by $(1 - 0.26)^5$ .

$V_0 = \frac{17420}{(1 - 0.26)^5}$

Step 4: Evaluate using a calculator.

$= \$78503.59621$

Step 5: Express the answer correct to the nearest whole dollar.

$= \$78504$

Step 6: Write the answer in words.

Angus paid $78,504 for the car.

Exam tip: When finding the purchase price, rearrange the formula to isolate $V_0$ before substituting values. This makes the calculation clearer.

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Worked Example 3: Calculating the percentage rate of depreciation

Madison bought a delivery van four years ago for $27,500. Using the declining-balance method for depreciation, she estimates its present value to be $8,107. What annual percentage rate of depreciation did she use? Answer to the nearest whole number.

Solution:

Step 1: Write the declining-balance depreciation formula.

$S = V_0(1 - r)^n$

Step 2: Substitute $S = 8107$ , $V_0 = 27500$ and $n = 4$ into the formula.

$8107 = 27500 \times (1 - r)^4$

Step 3: Make $(1 - r)^4$ the subject of the equation by dividing both sides by 27500.

$(1 - r)^4 = \frac{8107}{27500}$

Step 4: Take the fourth root of both sides to remove the power.

1 - r = \left(\frac{8107}{27500}\right)^{\frac{1}{4}}

Step 5: Rearrange to make $r$ the subject.

r = 1 - \left(\frac{8107}{27500}\right)^{\frac{1}{4}}

Step 6: Evaluate using a calculator.

$= 0.26314528$

Step 7: Express the answer correct to the nearest whole number (as a percentage).

$= 26\%$

Step 8: Write the answer in words.

Rate of depreciation is 26%.

Exam tip: When finding the rate, you'll need to use the root function on your calculator. The root you take matches the power in the equation (fourth root for power 4, fifth root for power 5, etc.).

Remember!

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Key Points to Remember:

Declining-balance depreciation reduces an asset's value by a fixed percentage each period, resulting in decreasing dollar amounts of depreciation over time.
The formula is $S = V_0(1 - r)^n$ , where $S$ is salvage value, $V_0$ is purchase price, $r$ is depreciation rate (as a decimal), and $n$ is number of time periods.
Always convert percentage rates to decimals before substituting into the formula (divide by 100).
This method works like compound interest in reverse – the percentage applies to the reduced value each period.
To find different unknowns, rearrange the formula: isolate $V_0$ to find purchase price, or solve for $r$ to find the depreciation rate.

Declining-Balance Depreciation (HSC SSCE Mathematics Standard): Revision Notes