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I'm not sure if I should use 'than' or 'from' in this sentence > "This transmitter extends three times signal range than the other one". Please advise. Thanks.

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    "extends three times signal range" is just wrong.
    – Hot Licks
    Oct 31 '20 at 2:48
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    'This transmitter extends the signal range achieved by the other one by a factor of 3.' Oct 31 '20 at 15:08
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As Hot Licks points out, this sentence doesn't really make sense, and changing "than" to "from" really wouldn't improve it. If I understand what you are trying to say, you are comparing the signal range of this transmitter to that transmitter, and this one has three times the range of that one. I'm not terribly fond of "extends" here. The use of "extends" implies that something active is happening in one to extend the range relative to the other, as opposed to the two simply having different specifications. I would say "This transmitter has three times the signal range of the other" or some variant of that.

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