How many years will it take for the price of gasoline, currently at $2.27 per gallon decreasing by $0.03 every two months, and oil at $0.45 per gallon increasing by $0.05 every six months, to be equal?

To determine how many years it will take for the price of gasoline and oil to be equal given their respective rates of change, we can start by establishing the equations for each over time.

The price of gasoline currently is $2.27 per gallon and it decreases by $0.03 every two months. In one year, there are 12 months, which means there are 6 two-month periods. Therefore, the price of gasoline after one year can be calculated as:

[ \text{Price of gasoline after } t \text{ years} = 2.27 - 0.03 \times 6t = 2.27 - 0.18t ]

The price of oil currently is $0.45 per gallon and it increases by $0.05 every six months. In one year, there are 2 six-month periods. Thus, the price of oil after one year will be:

[ \text{Price of oil after } t \text{ years} = 0.45 + 0.05 \times 2t = 0.45 + 0.10t ]

To find when the prices will be equal, we set the two expressions equal to each other:

[