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Dina's Dilemma on Tape

Dina was asked by her teacher to evaluate 999. Just before she plunged in, however, she decided to approximate the expected size of her final answer. She assumed that she could write 5 digits per inch along an adding machine tape. How long of a strip will she need?

And if you want to explore the situation further, repeat the problem for the following numbers: 9, 99, 9999, etc.


Source: Adapted from Oregon Mathematics Teacher, Sept. 2000, p. 44

Hint: Remember that 999 means [9(99)].


Solution Commentary: Let x = 999 = 9387420489

log(x) = 387420489 log(9)= 369693099.6...

Thus, x has about 10369693099.6 digits, or 369,693,100 digits

Which means (at 5 digits/inch), Dina will need 73,938,619.8 inches of tape, or 6,161,551.65 feet of tape, or about 1,167 miles of tape!