Answer:
The value of the truck would be $23,490.28 after 4 years.
Explanation:
Main concepts:
Concept 1. Percentages
Concept 2. Depreciation
Concept 3. Distributive property
Concept 4. Repeated multiplication as a power
Concept 1. Percentages
A percentage is effectively a unit. In the same way that 25¢ is the same as 0.25 US dollars, 15% is the same as 0.15 (unit-less number).
A common task is to find a percentage of a quantity. For example, finding 25% of 40. Mathematically, the word "of" here means multiply, so 25% of 40 means "25% * 40". But, since a percentage can be converted to a regular unit-less number, this is also the same as "0.25 * 40" which is 10. So, 25% of 40 is 10.
Concept 2. Depreciation
To depreciate means that the value of the thing decreases. In this question, the value of the truck depreciates by 15% (per year), so its value would decrease by whatever 15% of its current value is.
15% of $45,000 is 0.15 * $45,000 = $6,750 (this is the amount that the value decreases by during the first year)
To find the value after 1 year, we need to subtract this from the original value. We'll label this equation 1:
- Equation 1: $45,000 - $6,750 = $38,250
Common mistake: It is a common mistake for people to continue subtracting $6,750 for years 2, 3, and 4 thinking that this is subtracting another 15%. However, each year, the 15% depreciation is based on the new value of the truck, so during the second year, the truck started at a value of $38,250 (from the end of the first year), and 15% of $38,250 will be different from 15% of $45,000.
15% of $38,250 is 0.15 * $38,250 = $5,737.50 (this is the amount that the value decreases by)
To find the value after 2 year, we need to subtract this from the value at the end of year 1. We'll label this equation 2:
- Equation 2: $38,250 - $5,737.50 = $32,512.50
Concept 3. Distributive property
In Equation 1 above, the left side of the equation can be factored by using the distributive property in reverse, and factoring out "$45,000":
$45,000 - $6,750 is the same as
$45,000 * ( 1 - 0.15 ) = $38,250 -- Equation 1a
This expression illuminates what is happening to the value if we change the numbers inside the parentheses into percentages
$45,000 * ( 100% - 15% )
Notice that in the parentheses, we start with 100% (whatever 100% is), and we're subtracting 15%.
Similarly, in Equation 2, the left side of the equation can be factored by using the distributive property in reverse, and factoring out "$38,250":
$38,250 - $5,737.50 is the same as
$38,250 * ( 1 - 0.15 ) = $32,512.50 -- Equation 2a
Notice that in the parentheses, we again start with 100% (whatever 100% is), and we're subtracting 15%.
Concept 4. Repeated multiplication as a power
Quick recap from Equation 1a & Equation 2a:
For year 1, Equation 1a: $45,000 * ( 1 - 0.15 ) = $38,250
For year 2, Equation 2a: $38,250 * ( 1 - 0.15 ) = $32,512.50
However, note the equivalence in bold above. This means that we can replace the $38,250 in the "Year 2" equation with the entire left side of the equation from Year 1.
Thus, for Year 2, the equation become:
$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) = $32,512.50
While the value at the end of year 3 could be written
$32,512.50 * ( 1 - 0.15 )
notice that the left side again starts with the value that ended the year before. With another substitution, for Year 3, the equation becomes:
$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) = Value at end of year 3
Continuing the pattern, we need the value at the end of the year before, times ( 1 - 0.15 ) to get the value for the end of the following year.
$45,000 * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) * ( 1 - 0.15 ) = Value at end of year 4.
Notice that each year, we're multiplying by another factor of ( 1 - 0.15 )
Recall that multiplying by the same number or expression repeatedly can be simplified with an exponent. For example, five 2s multiplied together: 2*2*2*2*2 can be written as
.
Therefore, the value of the truck at the end of year 2 can be written as

The value of the truck at the end of year 3:

The value of the truck at the end of year 4:

Side note: the value of the truck at the end of any year "n" is

So, to answer the question, and find the value of the truck in 4 years, we just need to evaluate





Rounded to the nearest penny, the value of the truck would be $23,490.28 after 4 years.