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F1.2 · Analyse appreciation and depreciation

Learn to analyse appreciation and depreciation through clear examples and targeted practice.

Ontario Grade 9 Mathematics

Financial Literacy

How an asset’s value can rise or fall over time

A family buys a used bicycle for CAD 400. Later, it might be worth more if similar bicycles become harder to find, or less because it has been used. A rise in an asset’s value is called appreciation. A fall in its value is called depreciation. An asset is something of value that a person or group owns. In this lesson, you will learn how to describe and calculate these changes. You will use familiar ideas about percentages and multiplication, then apply them to values over time.

What you will learn

  • Explain appreciation and depreciation in everyday language.
  • Find a percentage increase or decrease in an asset’s value.
  • Describe how repeated percentage changes affect value over time.
  • Use a table and a simple equation to track a changing value.

1. What appreciation and depreciation mean

Appreciation means an asset increases in value. For example, a collectible item might rise in value if more people want it. Depreciation means an asset decreases in value. A bicycle might depreciate as it is used and becomes older. These words describe a change in value; they do not tell us why the change happened.
A percentage is a way to describe an amount in relation to 100. For example, 10% means 10 out of 100, or one tenth. To find 10% of CAD 400, divide CAD 400 by 10. The result is CAD 40. You can also change a percent to a decimal before multiplying: 10% is 0.10.
The starting value is the asset’s value before a change. The new value is its value after the change. If an asset appreciates, add the increase to its starting value. If it depreciates, subtract the decrease. The percentage change is usually based on the value just before that change. This matters when an asset changes more than once.
  • Appreciation is an increase in value.
  • Depreciation is a decrease in value.
  • A percentage change is calculated from the value before that change.

2. Finding a new value after one change

Suppose an asset’s value appreciates by 10%. First, find 10% of its starting value. Then add that amount to the starting value. For depreciation, find the percentage decrease and subtract it. The same calculation can be done in one multiplication.
For appreciation by a rate r, multiply the starting value by 1 plus the rate written as a decimal. For depreciation, multiply by 1 minus the rate written as a decimal. The number 1 represents the whole starting value. For example, if a value falls by 15%, then 85% remains. As a decimal, 85% is 0.85.
The letter V can stand for a value, and r can stand for the percentage rate as a decimal. These are just short names for numbers. Use the increase or decrease model that matches the situation. Check that the new value is greater than the starting value for appreciation and less for depreciation, unless the rate is zero.
Vnew=Vstart(1+r)V_{\text{new}}=V_{\text{start}}(1+r) for appreciation; Vnew=Vstart(1−r)V_{\text{new}}=V_{\text{start}}(1-r) for depreciation
  • For appreciation, find the increase and add it, or multiply by the remaining whole plus the rate.
  • For depreciation, find the decrease and subtract it, or multiply by the part that remains.
  • Write the rate as a decimal before using it in a multiplication.

3. Tracking changes over several years

When the same percentage change happens each year, apply it to the value at the start of that year. Do not keep using the original starting value for every year. After each change, the new value becomes the starting value for the next change.
A table helps show this clearly. The example below uses an asset worth CAD 200 that depreciates by 10% each year. Each year, 90% of the value at the start of that year remains. The table rounds each result to the nearest cent.
Notice that the value lost in the second year is smaller than the value lost in the first year. That is because 10% is being taken from a smaller value. Repeated depreciation does not mean subtracting the same dollar amount each year. Appreciation works the same way: a repeated percentage increase is found from the value at the start of each new period.
You can use the repeated-change rule one period at a time, or use a table to organize several periods. For a changing rate, use the rate given for each period rather than repeating one rate. Always identify the value at the start of the period before calculating.
Vnext=Vcurrent(1±r)V_{\text{next}}=V_{\text{current}}(1\pm r)
  • Each period’s percentage change applies to that period’s starting value.
  • The amount of the change in dollars can vary even when the percentage stays the same.
  • A table can help prevent using the wrong starting value.

4. Read the situation and check the result

Before calculating, decide whether the value is rising or falling. Look for words such as increases, grows, appreciates, decreases, loses value, or depreciates. Then identify the starting value and the percentage for that period.
Estimate the result before using exact arithmetic. If an asset depreciates by 20%, its new value should be about four fifths of the starting value. If it appreciates by 5%, the new value should be only a little higher. An estimate helps catch a mistaken sign or a misplaced decimal.
Keep the meaning of a result in mind. A calculated value is an amount of money, so include the currency and sensible rounding when the question asks for a money value. Do not confuse the dollar amount of a change with the percentage rate. A change of CAD 30 is an amount; 10% is a rate.
  • Identify the direction, starting value, and rate before calculating.
  • Estimate whether the answer should be higher or lower.
  • State the new value clearly and distinguish it from the amount of change.

A CAD 200 asset depreciating by 10% each year

TimeStarting value10% decreaseValue after decrease
StartCAD 200.00—CAD 200.00
After year 1CAD 200.00CAD 20.00CAD 180.00
After year 2CAD 180.00CAD 18.00CAD 162.00

Worked example

A device that depreciates

A tablet is worth CAD 600. Its value depreciates by 15% over one year. What is its value after one year?
  1. Identify the change
    The tablet depreciates, so its value decreases. The starting value is CAD 600, and the rate is 15%.
  2. Find the part that remains
    If 15% of the value is lost, then 85% remains. Write 85% as the decimal 0.85.
    100%−15%=85%;85%=0.85100\%-15\%=85\%;\quad 85\%=0.85
  3. Calculate the new value
    Multiply the starting value by the part that remains. This gives the value after the one-year decrease.
    600×0.85=510600\times0.85=510
  4. Check the result
    The tablet lost 15% of CAD 600, which is CAD 90. Subtracting CAD 90 gives CAD 510. The result is lower than the starting value, as depreciation requires.
    600×0.15=90;600−90=510600\times0.15=90;\quad 600-90=510
Answer: After one year, the tablet is worth CAD 510.
Check: The value is less than CAD 600, and the decrease is CAD 90, which is 15% of CAD 600.

Common mistakes and how to avoid them

Adding the percentage rate directly to the dollar value.
Correction: A percentage is a rate, not a dollar amount. Find the percentage of the starting value, then add or subtract that amount.
Using the original value to find every repeated percentage change.
Correction: For each new period, use the value at the start of that period. The value changes after each period.
Adding a depreciation rate instead of subtracting it.
Correction: Depreciation means a decrease. Find the portion that remains, or subtract the amount of the decrease.
Treating the percentage rate as the amount of money lost or gained.
Correction: First calculate that percentage of the starting value. The result is the money amount of the change.

Lesson summary

  • Appreciation means an asset’s value rises; depreciation means it falls.
  • For one change, calculate the percentage amount from the starting value and add or subtract it.
  • When the same rate repeats, apply it to the value at the start of each period.
  • Use a table and a quick estimate to check that the result makes sense.

Check your understanding

Question 1

An asset worth CAD 300 appreciates by 10%. What is its new value?
  1. CAD 30
  2. CAD 270
  3. CAD 330
  4. CAD 310
Show answer and explanation
CAD 330
Ten percent of CAD 300 is CAD 30. Add the increase to the starting value: CAD 300 + CAD 30 = CAD 330.

Question 2

An asset worth CAD 500 depreciates by 20%. What part of its starting value remains?
  1. 20%
  2. 80%
  3. 100%
  4. 120%
Show answer and explanation
80%
The lost part is 20%, so the remaining part is 100% − 20% = 80%.

Question 3

A value is CAD 100 at the start of a period. It depreciates by 10%. What starting value should be used to find a 10% decrease in the next period?
  1. CAD 100
  2. CAD 90
  3. CAD 10
  4. CAD 110
Show answer and explanation
CAD 90
After the first decrease, the value is CAD 90. The next period starts from that new value, so use CAD 90.

Key terms

Asset
Something of value that a person or group owns.
Appreciation
An increase in an asset’s value.
Depreciation
A decrease in an asset’s value.
Starting value
The value before a particular change takes place.
Percentage rate
A rate that describes an amount out of 100.

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