The price of gas has been steadily increasing 5.5% per year. If the cost of a gallon is now $2.95: A. What will be the cost in 5 years? B. What was the cost 12 years ago?

Steps to Solve

1. Identify the Compound Interest Formula

To calculate the future price of gas, we use the compound interest formula:

$$ A = P(1 + r)^t $$

where:

• ( A ) is the amount of money accumulated after n years, including interest.
• ( P ) is the principal amount (the initial amount of money).
• ( r ) is the annual interest rate (decimal).
• ( t ) is the time the money is invested for in years.

2. Calculate Future Price in 5 Years

Set ( P = 2.95 ), ( r = 0.055 ), and ( t = 5 ):

$$ A = 2.95 (1 + 0.055)^5 $$

Now, calculate:

$$ 1 + 0.055 = 1.055 $$

Then raise it to the 5th power:

$$ (1.055)^5 \approx 1.3025 $$

Now, multiply by the principal:

$$ A \approx 2.95 \times 1.3025 \approx 3.8434 $$

So, the future price in 5 years is approximately $3.84.

3. Calculate Past Price 12 Years Ago

To find the past price, we rearrange the compound interest formula:

$$ P = \frac{A}{(1 + r)^t} $$

We know ( A = 2.95 ), ( r = 0.055 ), and ( t = 12 ):

$$ P = \frac{2.95}{(1.055)^{12}} $$

First, calculate ( (1.055)^{12} ):

$$ (1.055)^{12} \approx 1.795856 $$

Now, find the past price:

$$ P \approx \frac{2.95}{1.795856} \approx 1.64 $$

So, the past price 12 years ago is approximately $1.64.