When drive-ins were at the height of their popularity in the late 1950s, some 4,000 existed in the United States, but today there are less than one-quarter that many.
there are less than one-quarter that many
there are fewer than one-quarter as many
there are fewer than one-quarter of that amount
the number is less than one-quarter the amount
it is less than one-quarter of that amount
"amount" is a construction that can only be used to refer to UNCOUNTABLE quantities -- i.e., continuous quantities / mass nouns, which are not divided into distinctly countable units.
for instance, you can have an "amount" of water, furniture, etc., but you CANNOT have an "amount" of people, items, etc.
this sentence is very clearly speaking about a countable quantity -- the number (not "amount") of drive-in theaters. since that's countable, any choice with "amount" can be eliminated immediately.
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