I've got a good one. My mother-in-law's 10 year-old dryer broke. Indicators turn on, but when you select start, the relay clicks, nothing turns on, and then an error code is displayed. Turns out, this is a common problem, where a relay will produce enough current to slowly melt one of its solder connections; it eventually blows out, leaving an easily identifiable scorch mark on the board. There was no continuity between the relay contact and the next point on the board. I was able to resolder (poorly likely) the connection and it appears that the circuit is restored. Instead of buying a new dryer or spending hundreds on repair, I MAY get her up and running for nothing more than my time and a little solder (and electricity)...fingers crossed on that one - another family member is waiting to install it for her, since we live hours away.
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I think the toughest decision I have to make, is when to replace old appliances. A dryer doesn't get much more efficient, it seems, so that's one easy decision when it comes to keeping a working dryer for forever or attempt repairs for as long as it's cheap. There are a bunch of things that just work or can be refurbed cheaply to give you decades of cheap service. I don't think that's true for all appliances, though.
My ten year-old fridge works just fine now (especially now that I've cleaned possibly 10 years of [censored] off the condenser coils and vents behind the fridge from the previous owner), but at some point, the purchase of a new fridge will pay for itself well within the expected life of the new fridge. The numbers are far from adding up now, of course, but technology keeps getting better, EVEN WITH refrigerant restrictions and useless features being added all the time to even modestly-priced fridges...
Another good one to replace more frequently than most of us think to, is window A/C units. Well-maintained units can last decades and still put out freezing cold air, but at a cost. That cost could be substantially higher than a newer Energy Star unit. We just got a new house and are bringing two 4 year-old Sharp units with us. The previous owners left 2 ancient units and one newer one. I'll have to investigate their energy usage. The good thing is, there are always people looking for cheap units, so getting rid of them will be easy and net me a bit of coin in the process.
Every appliance uses some amount of money to use. The more you use it, the more the potential savings for newer. There's something to be said for reliability, though. Why replace something that has given you decades of perfect service for some new-fangled version that might fail on you just outside of the warranty period, negating every bit of the savings you were hoping to achieve! I think that's why I tend to leave things be for a while and continue to do tons of research. The consequence of a bad decision or early adoption of new gear can be not only hundreds in capital expenditure, but also the risk of added inconvenience, loss and repair costs.