Capital costing — sum-of-years-digits (SYD) depreciation Let P be the equipment purchase cost, S the salvage value, n the life in years, and r the year number (r = 1 for the first year). What is the depreciation charge in the rth year using the SYD method?

Introduction / Context:
Depreciation allocates capital cost over useful life for accounting and economic evaluations. The sum-of-years-digits (SYD) method front-loads depreciation, reflecting higher early-year value loss common to process equipment.

Given Data / Assumptions:

• P: purchase (installed) cost of equipment.
• S: salvage value at end of life.
• n: total life in years.
• r: year index (r = 1, 2, …, n).

Concept / Approach:
SYD uses a weighting scheme proportional to remaining life at the start of each year. The denominator is the sum of the integers from 1 to n, i.e., n(n + 1)/2. The first-year weight is n, second year (n − 1), …, last year 1. Depreciation is the fraction of depreciable base (P − S) allocated by that weight.

Step-by-Step Solution:
Compute denominator (sum of digits): SYD = n(n + 1)/2.Weight for the rth year: w(r) = n − r + 1.Depreciation in year r: Dep(r) = [w(r)/SYD] * (P − S) = [(n − r + 1)/(n(n + 1)/2)] * (P − S).

Verification / Alternative check:
Check endpoints: r = 1 ⇒ factor = n/SYD (largest); r = n ⇒ factor = 1/SYD (smallest). Sum over r = 1…n returns (P − S), confirming the distribution is correct.

Why Other Options Are Wrong:

• (b) Uses ascending weight r, not the SYD decreasing weights.
• (c) Incorrect denominator and weight; does not sum to (P − S).
• (d) Straight-line method, not SYD.

Common Pitfalls:

• Mixing up straight-line and SYD methods.
• Using r instead of (n − r + 1) for the weight.

Final Answer:
Dep(r) = [(n - r + 1) / (n(n + 1)/2)] * (P - S)