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In British English, minutes is generally included when the time involves numbers other than 5, 10, 15, 20, 25.

For example, six past seven is unidiomatic in British English. The normal expression is six minutes past seven.

When the time involves numbers other than 5, 10, 15, 20, 25, is minutes generally included in American English?
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Not necessarily; it's six of one and half a dozen of the other.
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In my experience, Americans generally refrain from using the X past Y phrase except when discussing fifteen minute intervals. For instance, I typically only hear that it is either a quarter past X, half past X, or a quarter 'til X. A time like 7:06 would usually be articulated as, " Seven-oh-six."