Question

The cost of a machine depreciated by ₹ 900 during the second year and by ₹ 810 during the third year. Find:

1. the rate of depreciation.
2. the original cost of the machine.

Answer 1

Given:

• Depreciation on the machine during the second year = ₹ 900
• Depreciation on the machine during the third year = ₹ 810

Let:

• Original cost of the machine = (P)
• Rate of depreciation = (r%)

Step 1: Depreciation is on the reduced value year by year.

After the first year, value of machine = `P xx (1 - r/100)` 

After the second year, value = `P xx (1 - r/100)^2`

After the third year, value = `P xx (1 - r/100)^3`

Step 2: Calculate depreciation amounts in terms of (P) and (r).

Depreciation during the second year = Value at end of first year − Value at end of second year

`P(1 - r/100) - P(1 - r/100)^2 = 900`

Depreciation during the third year = Value at end of second year − Value at end of third year

`P(1 - r/100)^2 - P(1 - r/100)^3 = 810`

Step 3: Divide the two depreciation equations to find (r).

`(P(1 - r/100) - P(1 - r/100)^2)/(P(1 - r/100)^2 - P(1 - r/100)^3) = 900/810`

Simplify numerator and denominator:

`((1 - r/100)[1 - (1 - r/100)])/((1 - r/100)^2[1 - (1 - r/100)]) = 900/810`

The brackets `[1 - (1 - r/100)] = r/100`, so:

`((1 - r/100) xx r/100)/((1 - r/100)^2 xx r/100) = 900/810`

Cancel `r/100`:

`(1 - r/100)/((1 - r/100)^2) = 900/810`

Simplify:

`1/(1 - r/100) = 900/810`

`1/(1 - r/100) = 10/9`

Therefore,

`1 - r/100 = 9/10`

⇒ `r/100 = 1 - 9/10`

⇒ `r/100 = 1/10`

⇒ r = 10

Step 4: Find the original cost (P).

From Step 2’s depreciation formula for the second year:

`P(1 - 10/100) - P(1 - 10/100)^2 = 900`

P × 0.9 – P × 0.9² = 900

P(0.9 – 0.81) = 900

P × 0.9 = 900

`P = 900/0.09`

P = 10,000