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This question is actually TWO questions:
There are many many factors that will change these answers (how much you pay for fuel, what kind of car you're driving, where you're driving it, even what weather are you driving in, etc).
The #2 question is easier to answer. Lets take a specific car to work the numbers: a 2022 Toyota Corolla with the 1.8L engine with USA emissions. We'll assume we're driving on level ground in conditions matching the EPA fuel testing guidelines on the highway. This gives us 38MPG source
Lets also assume you're in New York where the average price of 87 octane gas is currently $3.34/gallon source
With our $3.34/gallon gas at 38MPG, that means $0.0879 (about 9 cents) a mile or $0.00001664 per foot.
So the answer to question #2 is: one penny of fuel for this car in this place would travel about 600ft
As to the answer for question #1:
"a report issued by the Aluminum Association, Inc. based on research by Ricardo. The chart below shows that for a small car with a 1.6-liter engine, reducing weight by five percent led to an increase in fuel economy of 2.1 percent on the EPA combined rating." source
So lets assume we get the same for our 1.8L Corolla. A 2.1% fuel economy would take our 38MPG to 38.798MPG.
With our $3.34/gallon gas at 38.798MPG, that means $0.086 (still about 9 cents a mile) or $0.00001664 per foot. One penny of fuel for this car in this place would travel about 614ft. So dropping 100lbs would make the car go 14ft more per gallon.
Modern pennies are copper plated zinc. There are approximately 181 copper plated zinc pennies in a pound. source So 18100 pennies is 100lbs. So 18100 pennies in the car would cost us 14ft of travel on our gallon of gasoline.
So the answer to your original question of "how long would I have to drive before that penny has cost me a penny of extra fuel?"
About 4,285 miles
Realistically the numbers are so small this is likely somewhat inaccurate because of rounding in various steps of calculation.
That's only 24 miles? That's crazy!
Double check the post. I had a math error that I edited my post for correction. The final answer I came up with was: About 4,285 miles