MATH 101 EXAM 2 STUDY GUIDE
apportionment - Answers - divide objects that are the same but each person or state
gets a different amount
state - Answers - a party having a stake in the apportionment
seat - Answers - the set of (M) identical, indivisible objects that are being divided among
the (N) states
population - Answers - a set of (N) positive numbers which denote the states' respective
populations
we write P = p1 + p2 + · · · + pN
- where P is the total population
- pN are the states' populations
apportionment method - Answers - a systematic procedure that guarantees a division of
M seats to N states
standard divisor (SD) - Answers - the ratio of population to seats
in other words,
SD = TOTAL POPULATION/NUMBER OF SEATS
standard quota (SQ) - Answers - (of a state) the exact fractional number of seats that
the state would get if fractions were allowed
we denote these q1, q2, ··· ,qN
SQ = STATE POPULATION/STANDARD DIVISOR
lower quota - Answers - the standard quota rounded down
**if you always allocate the lower quota, will you never allocate too many seats
upper quota - Answers - the standard quota rounded up
Hamilton's method - Answers - 1. allocate the Lower Quota
2. give the surplus seats to the state with the largest fractional parts until no more
surplus seats remain
**the maximum number of seats that could be allocated to a population using Hamilton's
method is one more than the lower quota
, Jefferson's method - Answers - 1. find a modified divisor (D) so that the lower quotas
add up to the number of seats
2. apportion to each state its modified lower quota
**removes issue of surplus seats
modified divisor - Answers - D < Standard Divisor
Try it (guess and check)
- adjust up if total seats assigned is more than total seats available
- adjust down if total seats assigned is still less than total seats available
Huntington-Hill method - Answers - uses a modified divisor, but uses the geometric
mean to determine a cut-off point
geometric mean - Answers - found by multiplying the numbers together and then taking
the square root of these numbers
quota rule - Answers - a state's apportionment should be either its upper quota or its
lower quota
**Jefferson's can violate upper quota
**Huntington-Hill can violate upper and lower quota
** Hamilton's doesn't violate this rule
Alabama paradox - Answers - a paradox in which the number of seats being
apportioned is increased and this results in a state getting apportioned fewer seats
**only Hamilton's can create this paradox
new states paradox - Answers - a paradox in which a new state with its fair share of
seats is added into the computation and this results in another state's apportionment
being affected in any way
**only Hamilton's can create this paradox
population paradox - Answers - a paradox in which a state's population increases and
this results in the state being apportioned fewer seats
**only Hamilton's can create this paradox
adjacent vertex - Answers - any two vertices connected by an edge
isolated vertex - Answers - no edges connecting the vertex (needs disconnected graph)
degree - Answers - number of edges meeting at the vertex
apportionment - Answers - divide objects that are the same but each person or state
gets a different amount
state - Answers - a party having a stake in the apportionment
seat - Answers - the set of (M) identical, indivisible objects that are being divided among
the (N) states
population - Answers - a set of (N) positive numbers which denote the states' respective
populations
we write P = p1 + p2 + · · · + pN
- where P is the total population
- pN are the states' populations
apportionment method - Answers - a systematic procedure that guarantees a division of
M seats to N states
standard divisor (SD) - Answers - the ratio of population to seats
in other words,
SD = TOTAL POPULATION/NUMBER OF SEATS
standard quota (SQ) - Answers - (of a state) the exact fractional number of seats that
the state would get if fractions were allowed
we denote these q1, q2, ··· ,qN
SQ = STATE POPULATION/STANDARD DIVISOR
lower quota - Answers - the standard quota rounded down
**if you always allocate the lower quota, will you never allocate too many seats
upper quota - Answers - the standard quota rounded up
Hamilton's method - Answers - 1. allocate the Lower Quota
2. give the surplus seats to the state with the largest fractional parts until no more
surplus seats remain
**the maximum number of seats that could be allocated to a population using Hamilton's
method is one more than the lower quota
, Jefferson's method - Answers - 1. find a modified divisor (D) so that the lower quotas
add up to the number of seats
2. apportion to each state its modified lower quota
**removes issue of surplus seats
modified divisor - Answers - D < Standard Divisor
Try it (guess and check)
- adjust up if total seats assigned is more than total seats available
- adjust down if total seats assigned is still less than total seats available
Huntington-Hill method - Answers - uses a modified divisor, but uses the geometric
mean to determine a cut-off point
geometric mean - Answers - found by multiplying the numbers together and then taking
the square root of these numbers
quota rule - Answers - a state's apportionment should be either its upper quota or its
lower quota
**Jefferson's can violate upper quota
**Huntington-Hill can violate upper and lower quota
** Hamilton's doesn't violate this rule
Alabama paradox - Answers - a paradox in which the number of seats being
apportioned is increased and this results in a state getting apportioned fewer seats
**only Hamilton's can create this paradox
new states paradox - Answers - a paradox in which a new state with its fair share of
seats is added into the computation and this results in another state's apportionment
being affected in any way
**only Hamilton's can create this paradox
population paradox - Answers - a paradox in which a state's population increases and
this results in the state being apportioned fewer seats
**only Hamilton's can create this paradox
adjacent vertex - Answers - any two vertices connected by an edge
isolated vertex - Answers - no edges connecting the vertex (needs disconnected graph)
degree - Answers - number of edges meeting at the vertex
