Q.A piece of machinery costing ₹10000 is expected to have a useful life of 4 years and a scrap value of zero. Find the annual depreciation charge using the sum-of-the-years digits method.

The key idea here is the Sum-of-the-Years' Digits (SYD) method, which accelerates depreciation compared to the straight-line method. The annual charge is a fraction of the depreciable amount, where the fraction changes each year.

Step 1: Calculate the sum of the years' digits.

For a 4-year life: $1 + 2 + 3 + 4 = 10$.

Step 2: Determine the depreciable amount.

Cost = ₹10000, scrap value = ₹0, so depreciable amount = ₹10000.

Step 3: Find the annual depreciation for each year.

The fraction for year $n$ is $\frac{\text{remaining life}}{\text{sum of digits}}$.

• Year 1: $\frac{4}{10} \times 10000 = ₹4000$
• Year 2: $\frac{3}{10} \times 10000 = ₹3000$
• Year 3: $\frac{2}{10} \times 10000 = ₹2000$
• Year 4: $\frac{1}{10} \times 10000 = ₹1000$

The question asks for "the annual depreciation charge" — this is not a single value, as the charge varies each year. If a single figure is required, it is the charge for the first year.

Why the Sum-of-the-Years Digits Method?

Straight-line depreciation spreads cost evenly, but many assets lose more value in their early years — a new car drops the most the moment you drive it off the lot. The sum-of-the-years digits (SYD) method captures this by applying a declining fraction to the depreciable amount each year.

The logic is simple: if an asset lasts $n$ years, you assign "weight" $n$ to the first year, $n-1$ to the second, and so on down to 1. The total weight is the sum of these digits: $1 + 2 + \dots + n = \frac{n(n+1)}{2}$. Each year's depreciation is then:

This naturally gives a larger charge early on and a smaller one later.

Step-by-Step Solution

1. Identify the given values.

Cost = ₹10,000, useful life $n = 4$ years, scrap value = ₹0. The depreciable amount is therefore ₹10,000 − ₹0 = ₹10,000.

2. Compute the sum of the years digits.

For $n = 4$:

Or using the formula: $\frac{4 \times 5}{2} = 10$. This is the denominator for all fractions.

3. Determine the fraction for each year.

The remaining life at the start of:

• Year 1: 4 years → fraction = $\frac{4}{10}$
• Year 2: 3 years → fraction = $\frac{3}{10}$
• Year 3: 2 years → fraction = $\frac{2}{10}$
• Year 4: 1 year → fraction = $\frac{1}{10}$

4. Calculate the annual depreciation charge.

Multiply each fraction by the depreciable amount ₹10,000:

• Year 1: $\frac{4}{10} \times 10,000 = ₹4,000$
• Year 2: $\frac{3}{10} \times 10,000 = ₹3,000$
• Year 3: $\frac{2}{10} \times 10,000 = ₹2,000$
• Year 4: $\frac{1}{10} \times 10,000 = ₹1,000$

5. Verify the total.

Sum of all charges: $4,000 + 3,000 + 2,000 + 1,000 = ₹10,000$, which exactly recovers the cost — as it must, since scrap value is zero.