Many 10-speed bikes are vintage bikes from the 1970s or 1980s, or kids' bikes. Ten-speed bikes incorporate front and rear derailleurs and often have old-style or cheap bottom brackets. Tune-ups on most bikes, not just 10-speeds, involve cleaning and checking the performance and positioning of major components, like wheels, brakes, steering and gears.
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Loosen your bike's brakes by slightly unscrewing the cable clamping bolts or using the quick cable releases. Loosen the nuts on your wheel axles with the appropriate combination wrench, usually a 15 mm wrench, or unclamp the quick release, then pull the wheels off the bike.
Clean the wheel rim, spokes and hub with a degreaser or soap and water and a rag. Pull a rag between each of your five rear sprockets to clean any stuck dirt and debris. Wipe down your entire bike frame with a wet, slightly soapy rag, then a wet clean one.
Spray degreaser on a damp rag and wipe grease and grime away from your bicycle's parts, including chain, brakes and derailleur.
Inspect both ends of each brake cable and shifter cable for fraying and replace any damaged or rusty cables, using the cable cutter where necessary to help remove the cable.
Rub any embedded debris off your brake pads with fine-grit sandpaper. If the pads are worn down extremely unevenly or past the wear line, replace them. Reinstall the wheels and hook the brakes back up, adjusting the brake cable tension to hold the brake pads about 1/8 inch from the rim.
Loosen the bolts on the brake pads with the appropriate sizes of combination wrench or Allen wrench. Center the brake pads between the upper and lower edges of the rim. The front end of the brake pad should lean in slightly toward the rim relative to the rear end of the brake pad.
Adjust the cable tension on your derailleurs, if necessary, for the chain to shift smoothly on the five rear gears and two front chain rings. Have a friend loosen the cable clamping bolt on the derailleur, then tighten it while you pull the cable tight with pliers; repeat as necessary to achieve the right tension. If your chain falls off the innermost cog on the front or rear, tighten the screw labeled L on the appropriate derailleur; if it falls off the outermost cog, tighten the screw labeled H.
Squeeze a few drops of light oil into each section of cable housing and on any springs or rotating parts on your brakes and derailleurs and move each part back and forth a few times to distribute the lubricant. Apply one drop of oil to each link of chain. Wipe off any excess oil, particularly any that got on the brake pads or wheel rims.
Inflate your tires to the recommended pressure with the tire pump. Take your bike for a test ride on level ground to check the soundness of your brake and shifter adjustments.