Depreciation and Appreciation 📉📈

Welcome, students! In this lesson, you will learn how values change over time when something loses value or gains value. This is important in real life and in IB Mathematics: Applications and Interpretation HL because it connects algebra, sequences, and financial modelling. By the end of this lesson, you should be able to explain what depreciation and appreciation mean, use formulas to calculate future or past values, and interpret results in context.

What are depreciation and appreciation?

Depreciation means a value decreases over time. A car, phone, or computer usually depreciates because it gets older, wears out, or becomes less desirable. Appreciation means a value increases over time. A house, piece of land, or some investments may appreciate because demand rises or the item becomes more valuable.

In mathematics, these situations are often modeled using percentage change. If an item changes by the same percentage each time period, the pattern is multiplicative, not additive. That means each new value is found by multiplying the previous value by a constant factor. This is why depreciation and appreciation are closely related to sequences and exponential growth or decay.

For example, if a laptop worth $1200$ depreciates by $15\%$ per year, after one year its value is $1200\times0.85=1020$. The factor $0.85$ means the laptop keeps $85\%$ of its value each year. If a house worth $300000$ appreciates by $4\%$ per year, its value after one year is $300000\times1.04=312000$. The factor $1.04$ means the house becomes $104\%$ of its previous value each year.

Key terminology and mathematical language

It is important to use the correct terms in IB Mathematics: Applications and Interpretation HL. The initial value is the starting amount, often written as $P$ or $V_0$. The rate of depreciation or rate of appreciation is the percentage change per period, written as $r$. The time period is the number of years, months, or other intervals, written as $n$ or $t$. The final value is the amount after the change.

For depreciation, the multiplier is usually $1-r$ when $r$ is written as a decimal. For appreciation, the multiplier is $1+r$.

A general model for repeated percentage change is

$$V=V_0(1+r)^n$$

when the value is increasing by a rate $r$. For depreciation, the model becomes

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

when the value decreases by a rate $r$.

Here, $V$ is the value after $n$ periods. These formulas are examples of exponential models because the variable $n$ is in the exponent. That means the value changes by a constant factor each period.

Depreciation: how value decreases over time

Depreciation is common in financial and business contexts. A company may buy a machine, and its book value decreases each year. A student might buy a scooter, and its resale value drops over time. In many exam questions, you are given a starting value and a depreciation rate and asked to find the value after several periods.

Suppose a camera costs $800$ and depreciates by $20\%$ per year. The decay factor is $0.80$. After $3$ years, its value is

$$V=800(0.80)^3$$

Calculate step by step:

$$V=800(0.512)=409.6$$

So the camera is worth $409.60$ after $3$ years.

You may also be asked to find the depreciation rate from two values. If a car is worth $25000$ now and $20000$ after one year, the multiplier is

$$\frac{20000}{25000}=0.8$$

So the car depreciated by $20\%$ in one year because $1-0.8=0.2$.

A useful real-world idea is that depreciation is often modeled using compound decrease. This means the loss each year is based on the updated value, not the original value. That is why the graph is curved, not a straight line. In technology-supported tasks, you may use a spreadsheet or graphing calculator to generate the sequence of values and compare the model with actual data.

Appreciation: how value increases over time

Appreciation happens when an asset becomes more valuable over time. Common examples include property prices, some savings or investments, and scarce collectibles. In school mathematics, appreciation helps you understand how money can grow when it is multiplied by a constant factor over each time period.

Suppose a house is worth $500000$ and appreciates by $3\%$ per year. The growth factor is $1.03$. After $5$ years, the value is

$$V=500000(1.03)^5$$

This gives approximately

$$V\approx500000(1.159274)\approx579637$$

So the house is worth about $579637$ after $5$ years.

If an investment grows from $1000$ to $1210$ in $2$ years, you can find the yearly growth factor by solving

$$1000(1+r)^2=1210$$

Divide both sides by $1000$:

$$(1+r)^2=1.21$$

Take the square root:

$$1+r=1.1$$

$$r=0.1$$

This means the investment appreciated by $10\%$ per year.

Appreciation is also a form of exponential growth. Because each period uses the new value, the increase compounds over time. That is why a small yearly rate can lead to a large long-term change. 📈

Working with formulas, sequences, and graphs

Depreciation and appreciation are strongly linked to sequences. If you list the value each year, you get a geometric sequence. A geometric sequence has a common ratio, which is the multiplier from one term to the next.

For depreciation, the common ratio is a number less than $1$, such as $0.92$ or $0.75$. For appreciation, the common ratio is greater than $1$, such as $1.05$ or $1.12$.

For example, if a machine depreciates from $10000$ by $10\%$ per year, the sequence is

$$10000,\ 9000,\ 8100,\ 7290,\dots$$

Each term is found by multiplying the previous term by $0.9$. The $n$th term can be written as

$$V_n=10000(0.9)^{n-1}$$

if $V_1=10000$ is the starting value.

Graphs help show the pattern visually. A depreciation graph slopes downward and gets flatter over time. An appreciation graph slopes upward and becomes steeper over time. The shape is not linear because the change is proportional to the current value. This is an important distinction in Number and Algebra: linear models change by a constant difference, while exponential models change by a constant ratio.

Solving problems and interpreting results

In IB Mathematics: Applications and Interpretation HL, you are not only expected to calculate values but also interpret them. That means you should explain what your answer means in context. For example, if a car depreciates to $40\%$ of its original value after $5$ years, you should say what that implies about resale value and ownership decisions.

A common type of question asks for the time needed to reach a certain value. Suppose a painting worth $2000$ appreciates by $6\%$ per year. When will it be worth more than $3000$? You solve

$$2000(1.06)^n>3000$$

Divide by $2000$:

$$(1.06)^n>1.5$$

Use logarithms to solve for $n$:

$$n>\frac{\log(1.5)}{\log(1.06)}$$

This gives approximately

$$n>6.99$$

So after $7$ years, the painting will be worth more than $3000$. This shows how algebra and technology can work together to solve realistic financial problems.

You may also need to compare two models. A real asset might depreciate quickly at first and then more slowly later. Another might appreciate only if the market is stable. In such cases, you should check whether the percentage rate is reasonable and whether the model matches the situation.

Conclusion

Depreciation and appreciation are essential ideas in Number and Algebra because they show how quantities change multiplicatively over time. Depreciation models decreasing value, while appreciation models increasing value. Both are described using exponential formulas and geometric sequences. These models are useful for cars, houses, electronics, investments, and business planning. For IB Mathematics: Applications and Interpretation HL, students, the key skills are recognizing the correct model, using the right formula, calculating values accurately, and explaining what the result means in context. 📘

Study Notes

Depreciation means a value decreases over time; appreciation means a value increases over time.
The main formulas are $V=V_0(1-r)^n$ for depreciation and $V=V_0(1+r)^n$ for appreciation.
The multiplier for depreciation is $1-r$, and the multiplier for appreciation is $1+r$.
These situations create geometric sequences because each term is found by multiplying by a common ratio.
Depreciation and appreciation are exponential models, not linear models.
A graph of depreciation curves downward, while a graph of appreciation curves upward.
Real examples include cars, phones, machines, houses, land, and investments.
In exam questions, always interpret your answer in context, not just as a number.
Technology such as a graphing calculator or spreadsheet can help model repeated percentage change and check results.
Understanding depreciation and appreciation helps connect algebra, sequences, and financial reasoning in IB Mathematics: Applications and Interpretation HL.