Problem 10 - Algebra 1

Question

You buy a used truck for \(\$ 20,000\). It depreciates at the rate of \(15 \%\) per year. Find the value of the truck in the given years. 3 years

Step-by-Step Solution

Answer

The value of the truck after 3 years will be approximately \$12,282.50

Step 1: Identify the depreciation rate and the initial value

The depreciation rate is 15% and the initial value of the truck is $20,000.

Step 2: Compute the value of the truck each year

To find the value of the truck each year, we subtract the depreciation from the value of the truck the previous year. The depreciation each year is 15% of the previous year’s value. So, the formula for calculating the price becomes: \( Value_{new} = Value_{old} - 0.15 * Value_{old} = 0.85 * Value_{old} \)

Step 3: Find the value after 3 years

Using the formula from step 2, we can calculate the value after 3 years. After first year the value will be $ \(Value_1 = 0.85 * 20000 = \$17,000\) . After second year the value will be $ \(Value_2 = 0.85 *17000 = \$14,450\). Finally, after the third year the value will be $ \(Value_3 = 0.85 * 14450 = \$12,282.50\)

Key Concepts

Depreciation Rate Calculation

When discussing truck depreciation, understanding how to calculate the depreciation rate is crucial. Depreciation is the reduction in an asset's value over time due to factors like wear and tear. Here, we focus on the formula used for calculating the yearly depreciation rate. If a truck depreciates at a rate of 15% per year, it means that each year, the value of the truck decreases by 15% of its current value.

The initial value of the truck is important as it's the base from which all depreciation calculations start. Let's assume your truck's initial value is $20,000. To calculate the depreciation for each year, you apply the rate percentage to the remaining value:

• For year 1: Multiply $20,000 by 15% (0.15). This gives you the amount that will be subtracted.
• The value at the start of year 2 is the initial amount minus the depreciation: $20,000 - $3,000 = $17,000.

This process continues each year, always applying the 15% to the new reduced value. This simple approach gives a clear picture of how each year's value is determined.

Value of Asset Over Time

Over time, the value of an asset, like a truck, decreases due to depreciation. It's important to understand how to track this value annually with the depreciation rate. Beginning with the initial purchase value, we calculate the diminished value after each year by applying our known depreciation rate:

• Start with an initial value of $20,000 in this case.
• For the first year, after subtracting the depreciation, the truck's value becomes $17,000.
• In the second year, the truck's new value is calculated by applying the 15% depreciation on $17,000, leading to a new value of $14,450.
• Finally, in the third year, applying 15% on $14,450 results in the truck's value being reduced to approximately $12,282.50.

This step-by-step reduction gives a logical sequence, showcasing how the asset's value is expected to decline over time. This method helps in financial planning, allowing anticipation of future asset worth.

Percent Decrease Formula

The percent decrease formula is handy in calculating depreciation and understanding how the truck's value diminishes over time. To apply this, use the formula:\[Value_{new} = Value_{old} - (Depreciation \ Rate \times Value_{old})\]Simplifying, it becomes:\[Value_{new} = (1 - 0.15) \times Value_{old}\]This means you multiply the current asset value by 85% (100% - 15%). The formula focuses on reducing the value by the depreciation percentage, directly showing how much gets deducted annually from the prior year's value.

This method is particularly beneficial for comparing year-to-year percent decreases and is a consistent way to project asset value reliably. It showcases how each year's depreciation directly impacts the asset, aiding in effective financial planning.