Spending $1,500 more today for a hybrid engine rather than a conventional gasoline engine will result in annual fuel savings of $300. How many years must these savings continue in order to justify the extra investment if money is worth 10% per year compounded monthly?

How do I solve this?

This is a time value of money question (Annuity specifically) where you are given a Present value, recurring payment and interest rate. You use these inputs to compute total duration or time;

You can solve this with a financial calculator and adjust the variables to monthly basis;

Present value; PV = - 1500

Monthly interest rate; I/Y = 10% / 12 = 0.833%

Recurring cashflow (converted to monthly CF) = 300/12 = 25

One-time cashflow; FV = 0

then compute total duration (in months); CPT N = 83.509 months

Convert the 83.5 months to years by dividing by 12;

83.5 / 12 = 6.96, so about 7 years.