A certain car model costs 21,000 euro now, and it devaluates 5% every year. How much will it cost in 6 years?

Question

A certain car brand estimates that a certain model, that now costs 21000 euro, will devaluate 5% per year.

How much will this car cost in 6 years?

I did:

$$21000 \cdot (1+.05)^6 \approx 28142$$

Now I remove the initial value from this one to get the compound devaluation

$$28142-21000 = 7142$$

Then I subtract this value from the initial value

$$21000-7142 = 13858$$

But my book says the solution is 15,437 euro.

What did I do wrong?

Answer 1

You're supposed to use $0.95^5\approx 0.7351$ which gives the correct number. You DECREASE by 5 percent so $100\%-5\%=0.95$

Answer 2

Devaluate means value will decrease. So price will be $(1-0.05)$ times previous year's price.

So, new value = $$21000 \cdot (1-0.05)^6 \approx 15436.93$$

Understood?

Answer 3

You're adding $5\%$ to the value each year when you multiply by $(1+.05)^6$. To subtract $5\%$ each year, use $(1-.05)^6$. It's not the same, because after the first year, you're taking $5\%$ from a smaller value each year. If it grows, you're adding $5\%$ to a larger value each year.

Answer 4

If the care loses $5\%$ value each year, then its value is $(100\%-5\%)=95\%$ of its value the previous year.

Answer 5

Don't rote the formula. It is very easy to derive it.

The sentence that "a certain model, that [in year t] costs [f(t)] euro, will devaluate 5% per year" can be formalized mathematically as:

$$f(t+1) = f(t) \cdot (100\% - 5\%)$$

Substitute $f(t+1)$ into $f(t)$, and we get:

\begin{align*} f(t+2) &= f(t+1) \cdot (100\% - 5\%)\\ &= f(t) \cdot (100\% - 5\%) \cdot (100\% - 5\%)\\ &= f(t) \cdot (100\% - 5\%)^2 \end{align*}

And so on and on, for 5 times, and voila, here's the formula for "[h]ow much will this car cost in 6 years". \begin{align*} &...\\ f(t+6) &= f(t) \cdot (100\% - 5\%)^6 \end{align*} If you forget the compound interest formula again, you can derive it back quickly, or use this method to check if your formula is correct.

Answer 6

A different way of organizing what was already said:

$ \$ 7142$ would be the total increment in the value if it was increasing year after year. But with the value increasing, the increment is each year greater. So this results greater than the correct total decrement, which is $$21000- 15437 = 5563$$

The number $15437$ is calculated directly, considering that each year the devaluation is represented by multiplication by $1-0.05$ (as this is less than $1$ the number effectively decreases, as we want).

Therefore after six years, $$21000 \cdot (1-.05)^6 \approx 15437$$