Question

A $17,900 truck is depreciated by 15.5% over 5 years. What's the truck worth at the end of 5 years?

Answer 1

If taken as written (depreciating 15.5% in total over the course of 5 years) = $15,125.50. If straight-line depreciation 15.5% each year, = $4,027.50. If declining balance, = $7,711.46.

Explanation:

If the van depreciates by 15.5% in total over 5 years, then we'd have:

`color(white)(00)17,900` - Original Cost
`ul(xx 15.5%` - Depreciation rate
`2,774.50` - Depreciation

Therefore,

`17,900-2,774.50=$15,125.50`

I'll note we can get there by "going the other way" - by multiplying the cost of the van by `1-"depreciation rate"`:

`17,900xx(1-15.5%)=17,900xx(84.5%)=$15,125.50`

However, depreciation rates are usually expressed as a per year measure and so I suspect the question is asking about a van that is depreciating by 15.5% per year over 5 years.

There are a couple of ways depreciation works. One way is straight-line depreciation, where we divide the original price and divide it into 15.5% chunks. If we take away 5 of those chunks, we get:

`17,900-5(2774.50)=17,900-13,872.50=$4,027.50`

Another way to do depreciation is with declining balance depreciation (which means that we apply the depreciation percentage to the value of the van each year - and so as the value decreases, so does the amount of depreciation). This gives:

Year 1: `17,900xx84.5%=15,125.50`

Year 2: `15,125.50xx84.5% = 12,781.05` (we round to the closest cent)

Year 3: `12,781.05xx84.5%=10,799.99`

Year 4: `10,799.99xx84.5%=9,125.99`

Year 5: `9,125.99xx84.5%=$7,711.46`

Answer 2

`$7,711.46`

Explanation:

Depreciation is usually regarded annually, because during the year the value drops from what it was at the start of the year.

Use the compound formula, but because the value is decreasing by `15.5%` we subtract. This means that at the end of a year, the van is only worth `84.5%` of the value at the start of the year.

`"Value" = P(1-r)^n`

`= 17,900(1-0.155)^5" "larr (15.5/100 = 0.155)`

`=17,900(0.845)^5`

`=$7,711.46`

Note that this is the same as:

`17,900 xx84.5/100 xx84.5/100 xx84.5/100 xx84.5/100xx84.5/100`

(Each year the value drops by `15.5%`)