How can a dollar bill be worth more than a quarter when the quarter is metal and the bill is paper?

The dollar bill represents an amount of gold in the us possession. Modern change is just copper.

Correction and amplificationModern currency is no longer backed with gold, but simply what's called the "faith and promise" of the government. A particular bill or coin can be exchanged for goods or services in the denomination shown because there's an implicit agreement among all (or most) citizens to recognize it as having that exchange value.

Remember, all US bills are the same size and most are similar in manufacture so there's very little intrinsic difference between a $1 bill and a $100 bill except insofar as people consider one exchangeable for 100 times as many widgets as the other one. Similarly, a dollar coin only contains about a dime's worth of metal but it's recognized by consumers, banks, stores, and vending machines as buying a dollar's worth of whatever is being sold.