I have a women's Mongoose hybrid bike that is older. The tires are dry rotted and I can't read the tire size info.
How do I determine what size replacements to buy?
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Tire sizes are extremely complicated because there are so many different measurement standards. See: http://www.sheldonbrown.com/tire-sizing.html
One strategy will be for you to ascertain what the common tire size of your bike is (more likely than not 700c for a hybrid bike of the 1990s) and order that (I’m assuming you’re ordering online) but that won’t work if someone replaced the rims or wheels on your bike and returning tires would be a pain.
Really, the easiest is to take one of the wheels to a local bike shop (not a bike department in a big box store) and ask them. They can also help you get white wall tires if you want the classic look. If you’re cheap, just get the one wheel done and order the companion online but I’ve found my LBS to be within $5-10 of online prices and it’s worth the convenience. Note: you’ll also need a new inner tube and you’ll likely also need some new rim tape. The bike shop can also ascertain how much corrosion there is and whether or not it’ll affect the new tire.
Note that many older bikes from the 1970s and prior have steel rims which have lousy braking performance.
A more direct way to determine the size is to use a measuring tape and measure the diameter of the rim in millimeters. If you are able to pump the tires, you can also measure the width. Put together these numbers form the ETRTO size of the tire. For example, 32 millimeters wide tire on rim with 622mm diameter becomes 32-622.
Don't try to measure inch sizes, as already explained in the other answer they do not refer to any physical dimension of tire or rim. If you have to buy in inch sizes, use a conversion table and make sure you can return the tires if you got the size wrong.