hmmm..... total WAG here... I'll guess a dime is about 1/2" diameter, and a quarter is about 1" diameter, and figure the grooves are about 1/32" apart, so my guesses are:

dime => (.5 * 3.14) * 32 = 48-50 grooves
quarter => (1.0 * 3.14) * 32 = 96-100 grooves

with the variation being whether they groove them based on a decimal or binary system. :)

100 on both (dime's are closer together and quarters more spread out). Because everything on our money is symbolic. 50 states X 2

Never thought about counting them! Now I'm wondering about other things I've never wondered about before...
How many squares of TP are there in a roll of Quilted Northern?
How many whiskers on a dog?
How many holes in a saltine cracker?

Party music's idea sounds good to me.

LOL Some of us can't even see the date on a coin.

Holy carp! I've never even though of that! I like Rob's answer so I'll take a riff on that.

I say there are 50 on each.

LOL Kim, I'm with you!

and I'll guess 125 on them. (I wonder if Canadian coins have the same number as the American ones)

and I'll guess 125 on them.

(I wonder if Canadian coins have the same number as the American ones. However with robb333's logic there would be 26 grooves based on 10 provinces and 3 territories)