Growth and Decay
Dave bought a new car 8 years ago for $8400. Tobuy a new car comparably equipped now would cost $12,500. Assuming a steady rate of increase, what was the yearly rate of inflation in car prices over the 8-year period?
Is there an equation and could someone post the answer and (if someone could post the steps)?
To find the yearly rate of inflation in car prices over the 8-year period, we can use the formula for exponential growth:
P = P₀ * (1 + r)^n
Where:
P = Future value (current price of the car)
P₀ = Present value (initial price of the car)
r = Rate of inflation (unknown)
n = Number of years
We are given that the initial price (P₀) of the car was $8400, and the current price (P) is $12,500. The number of years (n) is 8. We need to solve for the rate of inflation (r).
Substituting these values into the formula, we get:
$12,500 = $8400 * (1 + r)^8
Now we can solve for r.
Divide both sides of the equation by $8400:
(1 + r)^8 = $12,500 / $8400
Simplify the right side:
(1 + r)^8 ≈ 1.4881
Take the eighth root of both sides to isolate (1 + r):
1 + r ≈ ∛1.4881
Simplify the right side:
1 + r ≈ 1.1048
Subtract 1 from both sides to solve for r:
r ≈ 1.1048 - 1
r ≈ 0.1048
Finally, convert the decimal to a percentage:
r ≈ 10.48%
Therefore, the yearly rate of inflation in car prices over the 8-year period is approximately 10.48%.
To find the yearly rate of inflation in car prices over the 8-year period, we can use the concept of percentage increase.
The formula for calculating the percentage increase is as follows:
Percentage Increase = ((New Value - Old Value) / Old Value) * 100
In this case, the Old Value is $8400, and the New Value is $12,500. Substituting these values into the formula, we can calculate the percentage increase:
Percentage Increase = (($12,500 - $8400) / $8400) * 100
Percentage Increase = ($4600 / $8400) * 100
Percentage Increase = 0.5476 * 100
Percentage Increase ≈ 54.76%
Therefore, the yearly rate of inflation in car prices over the 8-year period is approximately 54.76%.
The steps to finding the solution are:
1. Subtract the old value from the new value.
2. Divide the result by the old value.
3. Multiply the result by 100 to convert it to a percentage.
Keep in mind that this calculation assumes a steady rate of increase over the 8-year period.
8400 + 8x = 12500
8x = 4100
x = $512.50 (amount of inflation per year)
Note that you cannot convert this to percent rate because it wouldn't be steady. For the first year, 512.5/8400 = 6.1%. For the second year, the car is going to cost 8912.5, and 512.5/8912.5 = 5.75%. And so on...