
Ed M. answered 12/21/15
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I agree with Ellen A. that your sentence requires an article before the singular countable noun rate, though I believe that alongside Ellen's suggested correction of a most speakers of English would also accept the definite article the here, though some might argue that because the sentence does not identify a specific rate--i.e., it varies--the indefinite article a would be better, that is, human hair grows at one of a number of possible rates.
You also include two versions of your task, namely "Find the mistake and correct it" and "Find the mistake then [sic] write it correctly," both of which entail ("the mistake . . . it") that there is only one mistake in the sentence. However, I would argue that besides the lack of an article before rate, another error in the sentence is the placement of the hyphen between one and half. In Standard Written English, to express the measurement 1/2 or 0.5 of an inch in words we usually write one half-inch, though as with a number of compounds beginning with half the presence of a hyphen between half and the following word does seem to vary (though at what "rate" I couldn't begin to tell you). Nevertheless, "one-half-inch" would always be incorrect.
As for correcting this error, one issue is that you've got a coordinate structure in your sentence, "one-half [sic] to one inch a month" which clearly implies "one half-inch to one inch a month." In this case, i.e., where you omit the first mention of the measurement unit inch and only give it explicitly with the second mention after the coordinator (in this case, to), you'd be left with "one half-" which, despite the unusual appearance of a single word terminating with a hyphen, is in fact standard; that is, the most formal "correction" to your sentence would be "Human hair grows at a rate of about one half- to one inch a month."
Anyway, the previous issue leads me to suspect that perhaps in giving your sentence here you simply neglected to type the little word a before rate which was there in your original source for the sentence, and if I'm right then the "one-half" would count as "the" one mistake in your sentence that needs correction.
Finally (and you were foolish enough to believe I was done), some English speakers might also argue that using a before month is also "wrong" because in this meaning, in reference to a rate, the preposition per should be used instead, i.e., . . . one inch per month. Such an insistence flies in the face of the facts of English usage, however, in that even in formal registers the indefinite article a is used freely before a number of singular countable nouns referring to various time periods, e.g., a day, a week, a year, in reference to rates and few objections are ever raised.