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I recently took my bike in for a service and the mechanic urged me to replace my chain, potentially at great cost (might need a new wheel). He said it might jam or pop off. He couldn't explain to me why my safety might be threatened but repeatedly asserted that it would be threatened.

So, why exactly is an old stretched chain a safety issue? How exactly might I be more likely to crash because of this?

A worn, stretched chain will accelerate wear on sprocket and chainring teeth. There will then be a greater chance that the chain will slip off of and override the sprocket teeth as you apply a power stroke through the crank. When this happens the pedal and your foot suddenly slips downward, which could throw you off the bike, especially if you were off the saddle and standing on the pedals.

A new chain is not usually regarded as a 'great cost'. They are not cheap but not expensive either. There's no reason you need a new wheel, unless there are other problems of course. You probably do need a new cassette though.

Whether you need to replace the chain and cassette depends on your situation, your bike and how you ride. If you have a quality bike, do a lot of miles and want an efficient drivetrain you absolutely should replace it. If you do a few casual miles occasionally and are not experiencing any problems you can probably get away with leaving it as it is.

As chains stretch, they wear the chainrings and rear cogs and the teeth on them start to become pointed and eventually don't grip the chain so well. The danger is that, when you push on the pedals, the chain slips from tooth to tooth on the chainring, which turns with much less resistance than you were expecting – you just have the friction of the chain slipping against the teeth, rather than the resistance of you driving the bike forwards. When the pedals suddenly turn much easier than you were expecting, you can very easily fall off the bike, especially if you were pushing hard, which is the time the chain is most likely to slip. Any time you fall off the bike while you're just cycling along in traffic is a potential under-a-truck moment. To some extent, you can mitigate this by cycling gently but there's always the risk that you try to pedal hard in some kind of emergency situation, forget that you can't do that, and come off.

Having said that, there is something a little confusing. Maybe you just misunderstood your mechanic, but there's no reason that you should require a new wheel, unless that's completely independent from your need for a new chain. However, you may well need new chainrings and new rear cogs: because the stretched chain wears them to match itself, they'll also skip with a new chain.

Assuming you're using fairly low-end components (because people who have higher-end components tend to know more about their bike than you do) I'd expect a new chain to cost about 10 dollars/pounds/euros ("about" means the currency unit doesn't matter much among those), new rear cogs to be about 25 and new chainrings maybe 30ish. Plus labour.