1 Dollar = 10 Dimes

1 Dime = 10 Pennies

10 Dimes / 10 Pennies = 1 <--- But this is correct. We got a whole number. ~~10 Dimes fits into 10 Pennies 1 time.~~ 10 fits into 10 1 time.

1 = 10 dimes/dollar

1 = 10 pennies/dime

10 dimes / (10 pennies/dime) = 1 dimes^2/pennies

The only reason you think this calculation makes sense is because both factors are 10.

If you had a different money system that was still based on whole numbers, it wouldn't work.

1 pound = 20 shillings

1 shilling = 12 pence

20 shillings / (12 pence/shilling) = (5/3)(shillings^2/pence)

Does this mean the old english money system was invalid?

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1 Yard = 5280 Feet

1 Foot = 12 inches

5280 Feet / 12 inches = 440 <--- This is correct as well. ~~We got a whole number. 12 inches can fit neatly into 5280 feet 440 times.~~ 12 fits neatly into 5280 440 times.

The 1 Foot = 12 inches is implicit in the above equation.

1 nautical mile = 6076.12 feet

1 foot = 12 inches

1 nautical mile / 12 (inches/foot) = 506.34 nautical mile * feet/inches

Does this mean nautical miles don't exist?

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I just divided unlike units and got a right answer

I can do that too:

The speed limit on I-5: 70 miles/hour

Number of days in a week: 7 days / week

70 miles/hour / 7 days/week = 10 miles*week/hour*days

It's a whole number, it has to mean something.

How about

7 days in a week

24 hours in a day

60 minutes in an hour

7 days / 60 minutes/hour = 7/60 (days*hours/minutes)

7 days / 24 hours / day = 7/24 (days^2/hours)

Does this mean weeks don't exist?