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To determine how many positive integers less than 100 have reciprocals with terminating decimal representation, we need to consider the prime factorization of each integer. Any positive integer whose prime factorization consists only of 2s and/or 5s will have a reciprocal with a terminating decimal representation. In the range of positive integers less than 100, there are 49 numbers that are powers of 2, 5, or their products (2^0, 2^1, 2^2, ..., 2^6, 5^0, 5^1, 5^2). Therefore, there are 49 positive integers less than 100 with reciprocals having terminating decimal representations.

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