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If there are vacant seats in the senate such that 3/5ths or 2/3rds of the duly-chosen and sworn senators is not a whole number, how is a supermajority rounded?
e.g. If there are only 99 senators, 3/5ths is 59.4, so do you need 59 or 60 votes to invoke cloture?
tl;dr Before 1975, two-thirds of senators "present and voting" were required for a supermajority, assuming a quorum is met (as few as 51 * (2/3) = 34 senators). After 1975, different percentages are required for different proposals. Two-thirds are needed for removal from office or rules changes (67 Senators normally), only three-fifths for cloture (60 Senators normally). Cloture doesn't count vacant seats, so if two senators are dead/removed (59 >= 98 * (3/5) = 58.8).
A supermajority is a requirement that more than a simple majority is needed to adopt a proposal. Since majorities and supermajorities are the minimum level of support required to adopt a proposal, you round up when finding the required number of Senators (normally 51 for a simple majority, 60 for a three-fifths supermajority, and 67 for a two-thirds supermajority).
A supermajority or a qualified majority is a requirement for a proposal to gain a specified greater level of support than a 50% simple majority.
Rule XXII allowed a supermajority for cloture to end debate. In 1975, the rules were modified from Senators "present and voting" to "duly chosen and sworn." This change was implement in 1975
"Is it the sense of the Senate that the debate shall be brought to a close?" And if that question shall be decided in the affirmative by three-fifths of the Senators duly chosen and sworn -- except on a measure or motion to amend the Senate rules, in which case the necessary affirmative vote shall be two-thirds of the Senators present and voting -- then said measure, motion, or other matter pending before the Senate, or the unfinished business, shall be the unfinished business to the exclusion of all other business until disposed of."
This change was invoked to make it easier to end debate in the Senate, and make filibustering weaker.
Filibusters were particularly useful to Southern senators who sought to block civil rights legislation, including anti-lynching legislation, until cloture was invoked after a 60 day filibuster against the Civil Right Act of 1964. In 1975, the Senate reduced the number of votes required for cloture from two-thirds to three-fifths, or 60 of the current one hundred senators.
Cloture votes don't count vacant seats when figuring 3/5ths, so you still need 60 with one vacancy, but only 59 with two vacancies (59 >= 98 * (3/5) = 58.8).
Invoking cloture usually requires a three-fifths vote of the entire Senate—“three-fifths of the Senators duly chosen and sworn.” Thus, if there is no more than one vacancy, 60 Senators must vote to invoke cloture. [...] Failing to vote on a cloture motion has the same effect as voting against the motion: it deprives the motion of one of the 60 votes needed to agree to it.
It seems your question can be answered without discussions of whether the supermajority is of those "present and voting" or of those "chosen and sworn." That has nothing to do with your question, because it concerns the size of the number whose supermajority is being calculated, while your question concerns the method of calculating a supermajority for any given number.
The thresholds can be calculated without rounding. The number of votes is then compared to the threshold with the appropriate comparative operation. For a supermajority, the vote must meet or exceed the threshold; for a majority, it must exceed it.
A supermajority must be at least the specified fraction. That is, it must be greater than or equal to the fraction. So, a 3/5 supermajority:
Similarly, a 2/3 supermajority:
By comparison, a majority must be greater than half, so a majority
A Senate Super-majority in 60 votes. It does not matter if there are only 61 senators on the floor you need 60 votes to invoke cloture. In the case of impeachment it requires 67 votes to convict. In the case of rounding you always round up because 66 is less than the required fractional 66.666666... votes required. This would be true even if it were 66.000000001 votes required 66 votes would not be enough.
There are other types of votes that just require a simple majority(or 3/5, 2/3, etc) which would only require the number of members voting be counted. A member could abstain as well which means that the member can count towards a quorum but would not count towards the total number of votes needed to pass.
