Nicholas
Bill, sorry to say this but you're not going to find a good bow and set of arrows for $45 or less (new). You can go to Dick's sporting goods to see if they have a kid's set for that price or less, which they will, but those are best for shooters who are 100% beginner and don't expect draw weight shooting power of more than about 15# or precision accuracy. I think you're best to try ebay or Craig's List to find someone dumping an old one - you might get lucky! The cheapest recurve of reasonable quality that I know is $150 (Martin Jaguar Take Down Recurve Bow) with arrows costing about $100 (all new).