Business Mathematics: Depreciation - Sum-Of-The-Years’-Digits Method

With this method, the years of the asset’s lifetime are labeled 1, 2, 3, and so on, and the depreciation amounts are based on a series of fractions that have the sum of the years’ digits as their common denominator. The greatest digit assigned to a year is used as the numerator for the first year, the next greatest digit for the second year, and so forth.

Example
Cost of machine $17,000; salvage value, $2,000; estimated life, 5 years.
The depreciable amount is the cost less any salvage value:
$17,000 - $2,000 = $15,000

To find the fraction of this amount that is to be written off each year, proceed as follows:
1.   Label the years 1, 2, 3, 4, and 5.
2.   Calculate the sum (S') of the years’ digits by adding the digits assigned in step 1:
5 = 1 + 2 + 3 + 4 + 5 = 15
3.   Convert the sum to a series of fractions by placing each digit assigned in step 1 in the numerator of each respective fraction and placing the sum of the digits found in step 2 in the denominator:

4.    Take the above series of fractions in reverse order as the depreciation rates. Thus,

For a machine that has a short life expectancy, such as the one in the previous example, the method outlined in step 2 for finding the sum of the years’ digits suffices.

However, for a machine that has a long life expectancy, it is much simpler to use the following formula:

where S is the sum of the years’ digits and N is the life expectancy.
For the machine in Example 7, the calculation for the sum of the years’ digits would be


Example
The life expectancy of a piece of equipment is estimated to be 30 yr. The sum of the years’ digits by the formula used in the preceding example would be

In this case, N – 30 yr. Therefore


Solved Problems

4.46   Find the sum of the years’ digits for the following life expectancies: (a) 5 years, (b) 10 years, (c) 8 years, (d) 20 years, and (e) 15 years.
 

Solution


4.47   Determine the first year’s depreciation fraction that would be used in the sum-of-the-years’- digits method of depreciation for the following estimated useful lives: (a) 4 years, (b) 12 years, (c) 6 years, (d) 18 years, and (e) 3 years.
 

Solution
Since N is the estimated life, it would also be the numerator of the first year’s fraction. S is the sum of the years’ digits, and as such would be the denominator. Therefore, the first year’s depreciation fraction may be expressed as N/S.
For each of the given cases, we need to find S first:


4.48   Using the sum-of-the-years’-digits method, find the first year’s depreciation for a snowplow that cost the town of Holiday $8,000 and has an estimated useful life of 15 years and a salvage value of $1,500.
 

Solution

* Note that for ease of calculation, 15/120 can be reduced to 1/8 and/or written as the decimal 0.125.

4.49  If on January 1 Cliver Transport purchased two vans costing $50,000 each for their moving and storage business, what is the depreciation on both after the first year? The vans have a trade-in value of $5,000 each and an estimated life of 10 years each. (Round the depreciation to the nearest dollar.)
 

Solution


4.50   ABC Lighting purchased display cases totaling $5,670. Their estimated useful life is 8 years, and their salvage value is $1,200. Using the sum-of-the-years’-digits method, determine the depreciation to the nearest cent for the first 2 years.
 

Solution

Depreciation fraction for year 1 = N/S = 8/36.

Recall from Example 7 that we take the sequence of fractions in reverse order for our calculations. Hence, the numerator of year 1 fraction is the last year of useful life (i.e., the estimated life) of the item. Since we are moving backward, the numerator of the next year’s fraction is the estimated life minus 1 year. This may be stated as follows:
 

Solution


4.51   A piano was purchased by Melody Disco for $6,750. Using the sum-of-the-years’-digits method, what is the depreciation to the nearest cent for the first 3 years if the piano has a salvage value of $960 and an estimated useful life of 25 years?
 

Solution

* See previous problem.
Because we are moving in reverse sequential order from the total years of estimated life, we subtract yet one more year from N to find the numerator of the depreciation fraction for year 3.


4.52   Using the sum-of-the-years’-digits method of depreciation, what is the book value (to the nearest cent) at the end of the first year on a piece of equipment that cost $20,650? The equipment has a salvage value of $3,650 and an estimated useful life of 50 years.
 

Solution


4.53   Using the sum-of-the-years’-digits method, determine the accumulated depreciation after 3 years on an item that cost $125,000 and has an estimated useful life of 75 years and a salvage value of $10,000. (Round the depreciation to the nearest dollar.)
 

Solution



4.54   Using the sum-of-the-years’-digits method, prepare a depreciation schedule for a pizza oven that was purchased for $3,000 and has an estimated useful life of 5 years and a salvage value of $900.
 

Solution


4.55   Using the sum-of-the-years’-digits method, prepare a depreciation schedule for the first 5 years for machinery that cost $75,000. The machinery has a salvage value of $6,000 and an estimated useful life of 40 years. (Round depreciation to the nearest cent.)
 

Solution


4.56  Use the sum-of-the-years’-digits method to find the depreciation (to the nearest cent) for the first year for each of the following:

Solution
(a) S = N(N + l)/2 = 6(6 + l)/2 = 21; depreciation fraction = N/S = 6/21

(b) S = N(N + l)/2 = 12(12 + l)/2 = 78; depreciation fraction = N/S = 12/78

(c) S = N(N + l)/2 = 25(25 + l)/2 = 325; depreciation fraction = N/S = 25/325

(d) S = N(N + l)/2 = 8(8 + l)/2 = 36; depreciation fraction = N/S = 8/36

* Depreciable amount = cost - salvage value

 

4.57   Using the sum-of-the-years’-digits method, prepare a depreciation schedule for the first 5 years for 40 beds purchased by Hansen Motel. The beds cost $200 each. The charge to have all of them delivered was $200. Their life expectancy is 15 years with no salvage value. (Round depreciation to the nearest dollar.)
 

Solution


4.58   Using the sum-of-the-years’-digits method, prepare a depreciation schedule for the first 4 years for an item that cost $56,400 and has a salvage value of $4,500 and an estimated useful life of 30 years. (Round the depreciation to the nearest cent.)
 

Solution


4.59   Use the sum-of-the-years’-digits method to determine the accumulated depreciation at the end of the second year on a backhoe purchased by Van Dyke Excavating, Inc. on April 1 for $8,500. The backhoe has an estimated useful life of 5 years and a trade-in value of $1,000. (Round the depreciation to the nearest dollar.)
 

Solution

Depreciation fraction:

Year 1

Portion applicable to first calendar year:
$2,500 X 9 mo/12 mo in a year = $1,875*
Year 2. Since the first 3 months of the second calendar year are still part of the equipment’s first year of useful life, the depreciation for these months is calculated on the basis of the first year’s fraction as follows:

Depreciation for remaining 9 months of second year:

Portion applicable to second calendar year:
$2,000 X 9 mo/12 mo in a year = $1,500*
The full depreciation for the second calendar year is therefore
$625 + $1,500 = $2,125
Accumulated depreciation at end of second year:
.$1,875 + $2,125 = $4,000

* Note that this calculation is the same as
Depreciation/yr 12 mo/yr = depreciation/mo
Depreciation/mo × applicable mo = depreciation for applicable portion of yr

4.60   Using the sum-of-the-years’-digits method, determine the depreciation for the second year on an item that cost $22,000, has a salvage value of $2,000, and has an estimated useful life of 10 years. It was purchased on September 24. (Round the depreciation to the nearest dollar.)
Solution

Depreciation fraction:

Since the first 9 months of the second calendar year are still part of the item’s first year of useful life, the depreciation for these months is calculated on the basis of the first year’s fraction as follows:

Portion applicable to second calendar year:
$3,636 × 9 mo/12 mo in yr = $2,727 for Jan. 1-Oct. 1 of 2nd calendar year

The last 3 months of the second calendar year are calculated on the basis of the second year’s depreciation fraction:

Portion applicable to second calendar year:
$3,273 × 3 mo/12 mo in year = $818 for Oct. 1-Dec. 31 of 2d calendar year

Depreciation for 2d year:
$2,727 + $818 = $3,545