Question: You are curious how much a car cost 10 years ago. Currently the average car price is $25,750 and the average inflation rate over the past ten years has been 2.2% per year.
How much did the average car cost 10 years ago rounded to the nearest dollar?
1. **State the problem:** We want to find the price of the car 10 years ago given the current price and the average annual inflation rate.
2. **Formula used:** The relationship between the past price $P_0$, current price $P$, inflation rate $r$, and time $t$ years is given by the compound interest formula:
$$P = P_0 (1 + r)^t$$
where:
- $P = 25750$
- $r = 0.022$ (2.2% as a decimal)
- $t = 10$
3. **Rearrange to find $P_0$:**
$$P_0 = \frac{P}{(1 + r)^t}$$
4. **Calculate the denominator:**
$$1 + r = 1 + 0.022 = 1.022$$
$$ (1.022)^{10} = 1.2434 \text{ (rounded to 4 decimal places)}$$
5. **Calculate $P_0$:**
$$P_0 = \frac{25750}{1.2434}$$
6. **Show cancellation step:**
$$P_0 = \frac{25750}{\cancel{1.2434}} \times \frac{1}{\cancel{1.2434}} = 20718.7$$
7. **Round to nearest dollar:**
$$P_0 \approx 20719$$
**Final answer:** The average car cost approximately $20719$ ten years ago.