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a) If the cost is depreciated at the rate of 12% per annum, the cost of a photocopy machine becomes Rs 61,952 after 2 years.

(i) If [tex]V[/tex] = initial price, [tex]T[/tex] = time, [tex]R[/tex] = rate of depreciation, and [tex]V_1[/tex] = price after [tex]T[/tex] years, write the relationship among [tex]V[/tex], [tex]T[/tex], [tex]R[/tex], and [tex]V_1[/tex].

(ii) Find the original price of the machine.

(iii) If the rate of compound depreciation for the first and second years were 10% and 15% respectively, what would be the price of the photocopy machine after 2 years?

Sagot :

GinnyAnsweridn
Sure! Let's go through each part of the question step-by-step:

a) Initial Problem Statement:
The cost of a photocopy machine depreciates at the rate of 12% per annum, and its value becomes Rs. 61,952 after 2 years.

### Part (i)

Given:
- [tex]\( V \)[/tex] = initial price of the photocopy machine
- [tex]\( V_1 \)[/tex] = price of the photocopy machine after [tex]\( T \)[/tex] years
- [tex]\( R \)[/tex] = annual rate of depreciation
- [tex]\( T \)[/tex] = time in years

Relationship among [tex]\( V \)[/tex], [tex]\( V_1 \)[/tex], [tex]\( T \)[/tex], and [tex]\( R \)[/tex]:
The price after depreciation for [tex]\( T \)[/tex] years can be described by the formula:
[tex]\[ V_1 = V \times (1 - R)^T \][/tex]

### Part (ii)
Finding the original price of the machine:

Given:
- [tex]\( V_1 \)[/tex] = Rs. 61,952 (price after 2 years)
- [tex]\( R \)[/tex] = 12% = 0.12 (annual rate of depreciation)
- [tex]\( T \)[/tex] = 2 years

Using the depreciation formula:
[tex]\[ V_1 = V \times (1 - R)^T \][/tex]

Solving for [tex]\( V \)[/tex]:
[tex]\[ V = \frac{V_1}{(1 - R)^T} \][/tex]
[tex]\[ V = \frac{61,952}{(1 - 0.12)^2} \][/tex]
[tex]\[ V = \frac{61,952}{(0.88)^2} \][/tex]
[tex]\[ V = 61,952 \div 0.7744 \][/tex]
[tex]\[ V = 80,000 \][/tex]

So, the original price of the photocopy machine is Rs. 80,000.

### Part (iii)
Finding the price after compound depreciation rates of 10% for the first year and 15% for the second year:

Given:
- Initial price [tex]\( V \)[/tex] = Rs. 80,000
- Depreciation rate for the first year = 10% = 0.10
- Depreciation rate for the second year = 15% = 0.15

First, calculate the price of the machine after the first year:
[tex]\[ V_1' = V \times (1 - 0.10) \][/tex]
[tex]\[ V_1' = 80,000 \times 0.90 \][/tex]
[tex]\[ V_1' = 72,000 \][/tex]

Next, calculate the price of the machine after the second year:
[tex]\[ V_2'' = V_1' \times (1 - 0.15) \][/tex]
[tex]\[ V_2'' = 72,000 \times 0.85 \][/tex]
[tex]\[ V_2'' = 61,200 \][/tex]

So, if the rate of depreciation was 10% for the first year and 15% for the second year, the price of the photocopy machine after 2 years would be Rs. 61,200.
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