Solution:

Given:
Original cost of the machine = ` 1,00,000
Ultimate scrap value = ` 30,000
Rate of depreciation = 20% (on reducing balance method)

To find:
Effective life of the machine

Calculation:

Step 1:
Depreciation rate = 20%
Therefore, the amount of depreciation for the first year = 20% of ` 1,00,000 = ` 20,000
So, the value of the machine at the end of the first year = ` 1,00,000 - ` 20,000 = ` 80,000

Step 2:
Depreciation rate remains the same i.e., 20%, but it will be calculated on the reduced value of the machine, i.e., ` 80,000
So, the amount of depreciation for the second year = 20% of ` 80,000 = ` 16,000
So, the value of the machine at the end of the second year = ` 80,000 - ` 16,000 = ` 64,000

Step 3:
Similarly, calculating the depreciation for the third year, we get:
Depreciation for the third year = 20% of ` 64,000 = ` 12,800
Value of the machine at the end of the third year = ` 64,000 - ` 12,800 = ` 51,200

Step 4:
We will continue this process until the value of the machine reaches its scrap value of ` 30,000.
Let's assume that the effective life of the machine is 'n' years. Then,

Value of the machine at the end of the nth year = Scrap value of the machine = ` 30,000

So, the depreciation for the nth year = Value of the machine at the beginning of the nth year - Scrap value
20% of Value of the machine at the beginning of the nth year = Value of the machine at the beginning of the nth year - ` 30,000
Value of the machine at the beginning of the nth year = ` 30,000 / 0.2 = ` 1,50,000

Therefore, the effective life of the machine = n years
Initial value of the machine = ` 1,00,000
Scrap value of the machine = ` 30,000
Depreciation rate = 20% on reducing balance method
Value of the machine at the end of the nth year = Scrap value of the machine = ` 30,000

Using the formula of the sum of an infinite geometric series, we can find the effective life of the machine as follows:

` 1,50,000 = ` 1,00,000 (1 - 0.2)^n / (1 - (1 - 0.2))
` 1,50,000 / ` 80,000 = 0.8^n
1.875 = 0.8^n
Taking log on both sides, we get:
n = log(1.875) / log(0.8) = 7.28 years (approx.)

Therefore, the effective life of the machine is approximately 7.28 years.

Conclusion:

The effective life of the machine is approximately 7.28 years. Depreciation rate of

30000 = 100000(1-20/100)^n
0.3 = (0.8)^n
n ~ 5.4 years (approximately)