Problem 30 - Algebra 2

Question

Construction An earthmover purchased for \(\$ 600,000\) loses 18\(\%\) of its value each year. What is the value of the earthmover after one year? After three years?

Step-by-Step Solution

Answer

The value of the earthmover after one year is \(\$ 492,000\) and after three years, it is \(\$ 330,820.80\).

Step 1: Calculate value decrease after one year

For the first year, calculate 18\(\%\) of \(\$ 600,000\). This can be done by multiplying \(\$ 600,000\) by \(\0.18\) to get \(\$108,000\). Subtract this amount from the original price of \(\$ 600,000\). This results in \(\$ 600,000 - \$ 108,000 = \$ 492,000\). So, the value of the earthmover after one year should be \(\$ 492,000\).

Step 2: Calculate value decrease after two years

For the second year, do the same-taking 18\(\%\) of the value from the first year, \(\$ 492,000\). Multiply \(\$ 492,000\) by \(\0.18\) to get \(\$ 88,560\). Subtract this amount from \(\$ 492,000\), which results in \(\$ 492,000 - \$ 88,560 = \$ 403,440\). Hence, the value of the earthmover after two years should be \(\$ 403,440\).

Step 3: Calculate value decrease after three years

For the third year, again take 18\(\%\) of the value from the second year, \(\$ 403,440\). Multiply \(\$ 403,440\) by \(\0.18\) to get \(\$ 72,619.2\). Subtract this value from \(\$ 403,440\), resulting in \(\$ 403,440 - \$ 72,619.2 = \$ 330,820.80\). Therefore, the value of the earthmover after three years should be approximately \(\$ 330,820.80\).

Key Concepts

Percentage Decrease

Understanding percentage decrease is essential in solving depreciation problems like the one about the earthmover. A percentage decrease refers to how much a value is reduced in terms of a percentage of its original amount. In the exercise, the earthmover loses 18% of its value annually.
This means we multiply the current value by 0.18 to find the amount to subtract each year. This process reflects how much value depreciates due to the stated percentage decrease. It's important to remember that percentage decrease does not become constant in absolute terms every year; it is always calculated from the current value at each step.

Depreciation

Depreciation is the gradual reduction in value of an asset over time. In our case, the earthmover's value decreases every year by a specific percentage. Depreciation accounts for wear and tear, age, obsolescence, and similar factors.
The exercise uses an exponential decay model, as the value each year is a percentage of the remaining value, not the original value. Depreciation is different from a straight-line method where the same amount is reduced each year regardless of the remaining value.

Value Calculation

To calculate the value of the earthmover after a certain period of time, you need to apply the percentage decrease year by year. Starting with the original purchase price of $600,000, we calculate the decreased value after each year by applying the 18% rate.

• Year 1: Subtract 18% of $600,000 from $600,000 to find the new value.
• Year 2: Apply 18% to the new value from Year 1, and subtract again.
• Year 3: Apply 18% to the Year 2 value and subtract to get Year 3 value.

It's crucial each year's calculation uses the previous year's value, maintaining the exponential decay model.

Step-By-Step Solution

Following a step-by-step approach helps simplify complex calculations with depreciation. This breakdown ensures clarity. Here's how you can solve the exercise with ease:

• **First Year:** Multiply $600,000 by 0.18, getting $108,000. Subtract this from $600,000 to get $492,000.
• **Second Year:** Multiply $492,000 by 0.18 to find $88,560. Subtract to get $403,440.
• **Third Year:** Multiply $403,440 by 0.18, which is $72,619.20. Subtract to end with $330,820.80.

Notice how each step builds off the last, reinforcing the concept of exponential depreciation with percentage decrease. Viewing each step clearly allows the realization that depreciation affects assets over time rather than instantly.