To calculate the value of the motor truck as of June 30, 2022, using the double-declining balance (DDB) method, let’s break down the process into steps.

Step 1: Gather Initial Information

1. Acquisition date: December 1, 2021.
2. Depreciation method: Double-declining balance (DDB).
3. Useful life: 5 years.
4. Residual (salvage) value: $15,000.
5. Initial cost: Not specified, but we will denote it as $C$.

Step 2: Determine Depreciation Rate

The double-declining balance rate is calculated as: $\text{Depreciation Rate} = \frac{2}{\text{Useful Life}} = \frac{2}{5} = 0.4 \text{ or } 40\%$

Step 3: Calculate Depreciation for 2021

Since the truck was acquired on December 1, 2021, we’ll only consider one month of depreciation for 2021 (December).

1. Depreciation for December 2021: $\text{Depreciation for December 2021} = C \times 40\% \times \frac{1}{12} = \frac{C \times 0.4}{12} = \frac{C}{30}$

2. Book value as of December 31, 2021: $\text{Book Value at End of 2021} = C - \frac{C}{30} = \frac{29C}{30}$

Step 4: Calculate Depreciation for First Half of 2022 (January to June)

The depreciation for each of these six months is based on the remaining book value as of December 31, 2021.

1. Depreciation from January to June 2022: $\text{Depreciation for 6 Months in 2022} = \frac{29C}{30} \times 40\% \times \frac{6}{12} = \frac{29C}{30} \times 0.4 \times 0.5 = \frac{29C}{150}$

2. Book value as of June 30, 2022: $\text{Book Value at June 30, 2022} = \frac{29C}{30} - \frac{29C}{150} = \frac{145C - 29C}{150} = \frac{116C}{150} = \frac{58C}{75}$

Step 5: Conclusion

Without the initial cost $C$, the exact value can’t be computed, but the formula for the book value on June 30, 2022, is: $\text{Book Value at June 30, 2022} = \frac{58}{75} \times C$ To find the exact amount, substitute $C$ with the actual acquisition cost of the truck.

Would you like more details on this approach, or is there a specific part you'd like me to clarify?

Here are five related questions for further exploration:

1. How would the calculation change if using straight-line depreciation instead?
2. What if the truck's useful life was revised to 4 years?
3. How would partial-year depreciation work if acquired mid-month?
4. What happens to depreciation calculations if the residual value changes?
5. How is double-declining balance adjusted if the book value reaches residual value early?

Tip: For double-declining balance, depreciation expense is higher in the earlier years, making it ideal for assets that lose value faster initially.