Video Transcript
The value of a brand-new car which
costs 498,300 Egyptian pounds decreases at a rate of 15 percent every year. Find the value of the car after
eight years, giving the answer to the nearest pound.
For this question, we’ve been given
some information about the starting value of a car. We’re told that its value decreases
by 15 percent every year. And we’re asked to find the value
of the car after eight years.
Let’s begin with the information
about the depreciation of the car; that’s the decrease in value over time. We’re told that the car loses 15
percent of its value every year. So, if we take the starting value
to be 100 percent, then we’re going to subtract 15 percent from this. We find that the new value of the
car each year is 85 percent of its value from the previous year. We can also divide this value by
100 to find the decimal equivalent of 85 percent is 0.85. In doing this, we have found that
the value of the car in a given year can be found by multiplying its value from the
previous year by 0.85. Note that the question also tells
us that this happens every year.
With this information, we can form
a pattern that allows us to represent the yearly values of the car as a geometric
sequence. Remember, a geometric sequence, or
a geometric progression, is one where each term is found by multiplying the previous
term by a fixed nonzero number called the common ratio. The 𝑛th term in a geometric
sequence can be represented as 𝑎 sub 𝑛. This 𝑛th term can be found by
multiplying the first term, 𝑎, by the common ratio, 𝑟, raised to the power of 𝑛
minus one.
In this case, we define the first
term, 𝑎, as the starting value of the car, which is 498,300 Egyptian pounds. We have previously discussed that
each year the value of the car can be found by multiplying the value from the
previous year by 0.85. Given that this multiplier is the
same every year, this is our common ratio 𝑟.
Finally, we need to find the value
of the car after eight years. This means that 𝑛 is equal to
eight. And we’ll be finding 𝑎 sub eight,
which is the eighth term in the sequence. We can find 𝑎 sub eight using the
formula for the 𝑛th term of a geometric sequence and substituting in our
values. This gives us that 𝑎 sub eight is
equal to 498,300 multiplied by 0.85 to the power of eight minus one, in other words,
0.85 to the power of seven. This is equal to 159,743.563 and so
on.
Remember, this question asks us to
give our answer to the nearest pound. If we consider our units, this
means we must round our answer to the nearest whole number, that is, 159,744. In doing this, we have found the
answer to the question. To the nearest pound, the value of
the car after eight years is 159,744 Egyptian pounds.
