Depreciation - Algebra-1

1. Fundamental Concepts

Depreciation refers to the process by which the value of assets (such as vehicles, equipment, real estate, etc.) gradually decreases over time due to wear and tear, aging, market changes, and other factors. It is a phenomenon of exponential decay, where the value of an asset decreases at a fixed proportion year by year (or in other time units).

Formula: The general formula for depreciation is: $V = A \cdot (1 - r)^t$ Where:
- V represents the value of the asset after time t (remaining value);
- A is the initial value of the asset (value at the time of purchase);
- r is the depreciation rate per period (expressed as a decimal, e.g., 5% is 0.05);
- t is the time, and its unit must be consistent with that of the depreciation rate r (such as years, months, etc.).
Key Requirement: The depreciation rate r and time t must share the same time unit (for example, if r is an annual depreciation rate, t should be in years).

Easy Level: A car has an initial value dollars, with an annual depreciation rate (i.e., 0.1). Find its value V after 3 years.
Solution: Substitute into the formula
Medium Level: A piece of equipment has an initial value dollars, with a semi-annual depreciation rate (i.e., 0.03). Find its value V after 2 years.
Solution: There are 4 semi-annual periods in 2 years, so . Then
Hard Level: A set of machines has an initial value dollars, with a monthly depreciation rate (i.e., 0.015). Find its value V after 1 year and 4 months.
Solution: 1 year and 4 months is 16 months in total, so . Then

Clarify Variable Definitions: Accurately distinguish between the initial value of the asset A, the depreciation rate r (pay attention to whether it is in decimal form), and time t. Ensure that the time units of r and t are consistent (e.g., if r is a monthly depreciation rate, t should be in months).
Convert Time Units: If the time includes mixed units such as years and months (e.g., 1 year and 4 months), it needs to be uniformly converted to the unit matching the depreciation rate (e.g., converted to 16 months).
Simplify Exponential Calculations: For high powers (e.g., ), you can use a calculator for step-by-step calculation or simplify the steps using the laws of exponential operations.
Verify the Reasonableness of Results: The depreciated value V must be less than the initial value A (due to the nature of exponential decay). If the result is abnormal, check whether the depreciation rate is correct (whether it is mistakenly written as a growth rate) and whether the time units match, etc.