Australian Curriculum 12 General Mathematics - 2020 Edition
6.05 Analysing depreciation
Lesson

Depreciation refers to the situation where items or investments lose value over time. We will consider two types of depreciation in this course:

• Straight line depreciation - the value reduces by the same amount every time period.
• Reducing balance depreciation - the value reduces by a percentage of the previous value every time period.

### Straight line depreciation

Straight line depreciation is a bit like simple interest in reverse because the principal is reduced by the same amount every time period. The graph showing the value at regular intervals will appear as a downward-sloping straight line. The slope of the line reflects the fixed quantity lost from the value in each period. Faster depreciation means a graph with a steeper slope.

The straight line method assumes the value of depreciation is constant per period.

The straight line graph above shows a $\$20000$$20000 initial value with an annual depreciation of \3000$$3000 p.a., the item reduces to a worth of $\$0$$0 after approximately 6.76.7 years. Straight line depreciation can also be modelled using an arithmetic sequence. The recurrence relation for the graph above would be V_{n+1}=V_n-3000Vn+1=Vn3000 where V_0=20000V0=20000 and V_nVnrepresents the value at the end of the nnth year. #### Worked examples ##### Example 1 A car purchased for \20000$$20000depreciates by $\$2500$$2500 per year. How much is it worth after 66 years?  \text{Value }Value == \20000-\2500\times6$$20000−$2500×6$=$=$\$5000$$5000 ##### Example 2 A car purchased for \15000$$15000 is worth$\$9000$$9000 after 88 years. By how much did it depreciate per year?  \text{Depreciation per year}Depreciation per year == \frac{\text{Value lost}}{\text{Number of years}}Value lostNumber of years​ == \frac{\15000-\9000}{8}$$15000−$90008​ $=$= $\$750$$750 Select the brand of calculator you use below to work through an example of using a calculator for problems involving straight line depreciation. Casio Classpad How to use the CASIO Classpad to complete the following tasks regarding sequences in a straight line depreciation context. A car that is initially purchased for \36000$$36000 depreciates by $\$2250$$2250 each year. 1. Using a recursive relationship, generate the value of the vehicle at the end of each year for 55 years. 2. Using an explicit rule, generate the value of the vehicle at the end of each year for 55 years. 3. Determine the value of the car at the end of 88 years. 4. When does this depreciation model predict the car is worthless? 5. Graph the value of the vehicle from when it was purchased to when it became worthless. TI Nspire How to use the TI Nspire to complete the following tasks regarding sequences in a straight line depreciation context. A car that is initially purchased for \36000$$36000 depreciates by $\$2250$$2250 each year. 1. Using a recursive relationship, generate the value of the vehicle at the end of each year for 55 years. 2. Using an explicit rule, generate the value of the vehicle at the end of each year for 55 years. 3. Determine the value of the car at the end of 88 years. 4. When does this depreciation model predict the car is worthless? 5. Graph the value of the vehicle from when it was purchased to when it became worthless. #### Practice questions ##### question 1 The graph shows the depreciation of a car's value over 4 years. 1. What is the initial value of the car? 2. By how much did the car depreciate each year ? 3. After how many years will the car be worth \14400$$14400 ?

4. What is the value of the car after 4 years ?

##### Question 2

A car is initially purchased for $\$24000$$24000 depreciates by \1700$$1700 each year.

1. Write a recurrence relation, $V_n$Vn, that gives the value of the car, in dollars, after $n$n years.

Write both parts (including for $V_0$V0) on the same line, separated by a comma.

2. Use the sequence facility of your calculator to determine the value of the car after $7$7 years.

3. After how many years will the value of the car first fall below $\$10100$$10100? Your answer should be a whole number. ##### Question 3 A boat, initially purchased for \49000$$49000, depreciates at a rate of $\$180$$180 for every 10001000 km of use. 1. Write a recurrence relation, V_nVn, that gives the value of the boat, in dollars, after nn thousand kilometres. Write both parts of the relation (including for V_0V0) on the same line, separated by a comma. 2. Use the sequence facility of your calculator to determine the value of the boat after it has travelled 1100011000 km. 3. Use the sequence facility of your calculator to determine after how many kilometres the boat will first fall below \47917$$47917 in value.

Your answer should be a multiple of $1000$1000 km.

### Reducing balance depreciation

In a similar way to how investments with compound interest increase by a percentage of the value at the start of a time period, assets that are subject to reducing-balance depreciation decrease in value by a percentage of the value at the start of each time period.

This is the more common form of depreciation. We will calculate this kind of depreciation using two methods:

• Using the reducing balance depreciation formula, similar to calculating compound interest.
• Using a geometric sequence to model the situation.

### Reducing balance depreciation formula

The formula is just slightly different from the compound interest formula. The difference is that we are reducing the value so we must multiply the principal by a number less than $1$1 each time. For example, reducing by $5%$5% is the same as multiplying by $100%-5%$100%5% or $95%$95% or $0.95$0.95. Therefore the formula has $1-r$1r in the bracket instead of $1+r$1+r. (Or we could in fact consider it the same formula with a negative rate).

Depreciation formula

$A=P\left(1-r\right)^n$A=P(1r)n

where: $P$P is the principal (or initial) amount

$r$r is the depreciation rate per time period

$n$n is the number of time periods

$A$A is the value of the item after being depreciated

Note: The amount an item is worth after depreciation is also called the expected value , book value or residual value.

#### Worked examples

##### Example 3

Kathleen deposited $\$6500$$6500 into a new superannuation account. This amount decreased by 2%2% each year for 33 consecutive years. What was the value of her superannuation after 33 years? Give your answer to 22 decimal places if necessary. Think: How do we substitute these values into the depreciation formula? Do:  AA == P\left(1-r\right)^nP(1−r)n == 6500\times\left(1-0.02\right)^36500×(1−0.02)3 == 6500\times0.98^36500×0.983 Decreasing by 2%2% is the same as finding 98%98% == \6117.75$$6117.75

Select the brand of calculator you use below to work through an example of using a calculator for problems involving reduce balance depreciation using the formula and financial application.

Casio Classpad

How to use the CASIO Classpad to complete the following tasks involving reducing balance depreciation using the formula and the inbuilt financial solver.

A new car purchased for $\$28000$$28000 depreciates at a rate rr each year. 1. If the value of the car has reduced to \20412$$20412 after $3$3 years, find the value of $r$r as a percentage.

2. Assuming the rate of depreciation remains constant, how much can the car be sold for after $10$10 years? Give your answer to the nearest dollar.

3. The owner of the car wants to sell the vehicle before its value falls below $\$12000$$12000. According to the model, by the end of which year should he sell the vehicle? TI Nspire How to use the TI Nspire to complete the following tasks involving reducing balance depreciation using the formula and the inbuilt financial solver. A new car purchased for \28000$$28000 depreciates at a rate $r$r each year.

1. If the value of the car has reduced to $\$20412$$20412 after 33 years, find the value of rr as a percentage. 2. Assuming the rate of depreciation remains constant, how much can the car be sold for after 1010 years? Give your answer to the nearest dollar. 3. The owner of the car wants to sell the vehicle before its value falls below \12000$$12000. According to the model, by the end of which year should he sell the vehicle?

#### Practice questions

##### Question 4

A laptop currently worth $\$1719$$1719 was purchased 44 years ago for \2000$$2000.

What was the annual depreciation rate, $R$R?

Han's share portfolio of $\$83000$$83000 fell 14%14% per month for the first 44 months of the Global Financial Crisis and then 3%3% per month for the 55 months after that. What was the value of his portfolio after 99 months? Write your answer to the nearest cent. ##### Question 6 A dishwasher selling for \800$$800, depreciates at $11%$11% p.a. .

1. What is the percentage of the original value that will remain after 1 year?

2. What is the percentage of the original value that will remain after 2 years?

3. What is the percentage of the original value that will remain after 3 years?

4. How many full years will it take for the dishwasher to lose half its original value?

5. How many years will it take for the dishwasher to lose $90%$90% of its original value?

### Depreciation modelled as a geometric sequence

Reducing balance depreciation is where the value at the start of each year is multiplied by a constant rate of depreciation. Therefore we can solve depreciation problems using the geometric sequence forms, where the common ratio will always be less than $1$1. Notice the explicit rule is the same as the depreciation formula.

Reduce balance depreciation as a geometric sequence

For an item with initial value, $P$P, at a depreciation rate of $r$r per period, the sequence of the value of the item over time forms a geometric sequence with a starting value of $P$P and a common ratio of $(1-r)$(1r).

The sequence which generates the value, $V_n$Vn, of the item at the end of each depreciation period is:

• Recursive form:

$V_n=V_{n-1}\times(1-r)$Vn=Vn1×(1r), where $V_0=P$V0=P

• Explicit form:

$V_n=P(1-r)^n$Vn=P(1r)n

#### Worked example

##### Example 4

Danielle buys a car for $\$15000$$15000 and is told to expect it to depreciate at a rate of x%x% p.a. The progression of the depreciation is shown in the table below. Year Car value at start of year () Depreciation () Car value at end of year () 1 1500015000 22502250 1275012750 2 1275012750 1912.501912.50 10837.5010837.50 3 10837.5010837.50 1625.631625.63 9211.889211.88 (a) Determine x%x%, the depreciation rate of the car. Think: We can use any of the table rows to do this. Use the formula \text{Depreciation rate}=\frac{\text{Depreciation in year n}}{\text{Value at start of year n}}\times100%Depreciation rate=Depreciation in year nValue at start of year n×100% Do: Using the values from Year 1 we find:  Rate == \frac{2250}{15000}225015000​ == 0.150.15 == 15%15% Therefore, the depreciation rate is 15%15% p.a. Reflect: We can check we obtain the same result using the values from Year 2 or Year 3. (b) Write a recurrence relation for the investment. Think: Each time we are decreasing by 15%15%, which means multiplying by 0.850.85 as (100%-15%=85%100%15%=85%). We want n=1n=1 to show the value at the end of year 11, therefore let V_0=V0= initial value. Do: Write the rule V_{n+1}=V_n\times0.85Vn+1=Vn×0.85, where V_0=15000V0=15000 Or: Write the rule V_n=V_{n-1}\times0.85Vn=Vn1×0.85, where V_0=15000V0=15000 Select the brand of calculator you use below to work through an example of using a calculator for problems involving sequences and reduce balance depreciation. Casio Classpad How to use the CASIO Classpad to complete the following tasks involving sequences in a reducing balance depreciation context. A boat, originally purchased for \12000$$12000, depreciates at $15%$15% per annum.

1. Using a recursive relationship, generate the value of the boat at the end of each year for the first $5$5 years.

2. Using an explicit rule, generate the value of the boat at the end of each year for the first $5$5 years.

3. Find the value of the boat at the end of $8$8 years.

4. At the end of which year does the value of the boat first fall below $\$4000$$4000. 5. Sketch a graph of the value of the boat over the first ten years. TI Nspire How to use the TI Nspire to complete the following tasks involving sequences in a reducing balance depreciation context. A boat, originally purchased for \12000$$12000, depreciates at $15%$15% per annum.

1. Using a recursive relationship, generate the value of the boat at the end of each year for the first $5$5 years.

2. Using an explicit rule, generate the value of the boat at the end of each year for the first $5$5 years.

3. Find the value of the boat at the end of $8$8 years.

##### question 8

After one year, the value of a company’s machinery had decreased by $\$16020$$16020 from \89000$$89000. The value of the machinery depreciates by a constant percentage each year.

1. At what rate did the machinery depreciate in the first year?

2. What will the machinery be worth at the end of the second year?

3. Write a recurrence relation, $V_n$Vn, that gives the value of the machinery at the end of year $n$n.

Write both parts of the relation (including for $V_0$V0) on the same line, separated by a comma.

4. The company bought this machinery at the end of $2011$2011. When the value of the machinery falls below $\$30003000, they will invest in new machinery. In which year will this occur?

### Outcomes

#### ACMGM070

use arithmetic sequences to model and analyse practical situations involving linear growth or decay; for example, analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, or calculating the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation

#### ACMGM074

use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate

#### ACMGM096

with the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value