- WELCOME TO A LESSON
ON JEFFERSON'S METHOD 
OF APPORTIONMENT.
AFTER WASHINGTON VETOED 
HAMILTON'S METHOD,
THOMAS JEFFERSON PROPOSED 
A NEW METHOD
CALLED JEFFERSON'S METHOD.
IT WAS USING CONGRESS 
FROM 1791 THROUGH 1842.
THE METHOD DOES TEND 
TO FAVOR LARGE STATES,
AND JEFFERSON HAPPENED TO LIVE 
IN VIRGINIA,
THE LARGEST STATE AT THE TIME.
JEFFERSON'S METHOD DIFFERS 
FROM HAMILTON'S METHOD
ON HOW TO RESOLVE A SITUATION
WHEN THE LOWER QUOTA 
OR INITIAL QUOTA
IS LESS THAN THE ACTUAL SEATS 
AVAILABLE.
SO THE FIRST THREE STEPS 
OF JEFFERSON'S METHOD
IS THE SAME 
AS HAMILTON'S METHOD.
STEP ONE DETERMINE 
HOW MANY PEOPLE
EACH REPRESENTATIVE 
SHOULD REPRESENT.
WE DO THIS BY DIVIDING THE TOTAL 
POPULATION OF ALL THE STATES
BY THE TOTAL NUMBER 
OF REPRESENTATIVES.
THIS ANSWER IS CALLED 
THE STANDARD DEVISOR
OR JUST A DEVISOR.
STEP TWO WE DIVIDE EACH STATE'S 
POPULATION BY THE DIVISOR
TO DETERMINE HOW MANY 
REPRESENTATIVES IT SHOULD HAVE.
WE RECORD THIS ANSWER 
TO SEVERAL DECIMAL PLACES
AND THIS ANSWER IS CALLED 
THE QUOTA.
STEP THREE WE CUT OFF THE 
DECIMAL PART OF ALL THE QUOTAS.
THESE VALUES ARE THE LOWER 
QUOTAS OR INITIAL APPORTIONMENT.
WE ADD THESE WHOLE NUMBERS.
THIS ANSWER WILL ALWAYS BE 
LESS THAN OR EQUAL TO
THE TOTAL NUMBER 
OF REPRESENTATIVES.
IF IT'S EQUAL TO THE TOTAL 
NUMBER OF REPRESENTATIVES
WE WOULD BE DONE.
BUT IF IT'S NOT 
THEN WE GO TO STEP FOUR.
STEP FOUR SAYS IF THE TOTAL 
FROM STEP THREE
IS LESS THAN THE TOTAL NUMBER 
OF REPRESENTATIVES
WE ACTUALLY REDUCE THE DEVISOR
AND RECALCULATE THE QUOTA 
AND INITIAL ALLOCATION.
SO IF WE HAVE LEFT OVER 
REPRESENTATIVES
WE ACTUALLY MODIFY THE DEVISOR
RATHER THAN USING THE DECIMAL 
PARTS OF THE QUOTAS
AS WE DO 
WHEN USING HAMILTON'S RULE.
SO IF WE HAVE LEFT OVER 
REPRESENTATIVES
WE ACTUALLY CHANGE THE DEVISOR.
REMEMBER IN HAMILTON'S RULE
WE RELY ON THE DECIMAL PARTS 
OF THE QUOTA.
SO FOR JEFFERSON'S METHOD 
WE REDUCE THE DEVISOR
AND RECALCULATE THE QUOTA 
AND ALLOCATION.
WE CONTINUE DOING THIS 
UNTIL THE TOTAL FROM STEP THREE
IS EQUAL TO THE TOTAL NUMBER 
OF REPRESENTATIVES.
THE DEVISOR WE END UP USING 
IS CALLED THE MODIFIED DEVISOR
OR ADJUSTED DEVISOR.
SO LET'S LOOK AT AN EXAMPLE.
A COLLEGE OFFERS TUTORING 
IN MATH, ENGLISH, CHEMISTRY,
AND BIOLOGY.
THE NUMBER OF STUDENTS ENROLLED 
IN EACH SUBJECT IS LISTED BELOW.
IF THE COLLEGE CAN ONLY AFFORD 
TO HIRE 21 TUTORS,
DETERMINE HOW MANY TUTORS SHOULD 
BE ASSIGNED TO EACH SUBJECT
USING JEFFERSON'S METHOD.
SO THE FIRST STEP IS TO FIND 
THE TOTAL ENROLLMENT,
WHICH WE SEE HERE IS 890.
AND SINCE WE HAVE 21 TUTORS 
TO ALLOCATE,
WE FIND THE STANDARD DEVISOR BY 
TAKING 890 AND DIVIDING BY 21.
SO THE STANDARD DEVISOR 
IS APPROXIMATELY 42.3810.
AND NOW TO FIND THE QUOTA 
FOR EACH SUBJECT,
WE TAKE THE ENROLLMENT 
FOR EACH SUBJECT
AND DIVIDE BY OUR 
STANDARD DEVISOR.
LET'S GO AHEAD AND SHOW 
A COUPLE OF THESE.
SO TO FIND THE QUOTA FOR MATH
WE WOULD HAVE 360 
DIVIDED BY 42.3810.
TO THREE DECIMAL PLACES
THE QUOTA WOULD BE APPROXIMATELY 
8.494.
THE QUOTA FOR ENGLISH WOULD BE 
315 DIVIDED BY 42.3810,
WHICH WOULD BE APPROXIMATELY 
7.433.
WE WOULD DO THE SAME 
FOR CHEMISTRY AND BIOLOGY.
SO HERE WE SEE THE QUOTAS 
FOR THE FOUR DISCIPLINES.
AND NOW TO FIND THE INITIAL 
APPORTIONMENT
OR INITIAL ALLOCATION,
REMEMBER WE DROP THE DECIMAL 
PART OF THE QUOTA.
SO MATH GETS 8, ENGLISH GETS 7, 
CHEMISTRY GETS 3,
AND BIOLOGY GETS 1.
BUT NOTICE HOW USING 
THE INITIAL APPORTIONMENT,
NOTICE HOW THE TOTAL IS 19 AND 
WE HAVE A TOTAL OF 21 TUTORS.
SO NOW WE'RE GOING TO MODIFY 
THE STANDARD DEVISOR
OR REDUCE THE STANDARD DEVISOR,
RECALCULATE THESE QUOTAS
UNTIL WE FIND AN INITIAL 
APPORTIONMENT HERE
THAT DOES SUM TO 21.
SO LET'S GO AHEAD 
AND REDUCE THIS TO LET'S SAY 42
AND THEN RECALCULATE 
THESE QUOTAS.
SO IF WE USE AND MODIFY 
DEVISOR 42,
NOW WE'RE GOING TO TAKE THE 
ENROLLMENT OF EACH SUBJECT
AND DIVIDE BY 42.
AND LET'S GO AHEAD AND SHOW 
A COUPLE OF THESE.
SO, AGAIN, FOR MATH 
WITH THE MODIFIED DEVISOR
WE'D HAVE 360 DIVIDED BY 42,
WHICH IS APPROXIMATELY 8.571.
FOR ENGLISH WE'D HAVE 315 
DIVIDED BY 42,
WHICH IS EQUAL TO 7.5.
I THINK YOU GET THE IDEA,
BUT NOTICE HOW IF WE CUT OFF 
THE DECIMAL PARTS OF THE QUOTA,
NOTICE HOW WE HAVE 8, 7, 3, 1.
BUT STILL NOTICE HOW THE TOTAL 
HERE IS 19 NOT 21.
SO NOW WE NEED TO REDUCE 
THE DEVISOR AGAIN.
LET'S GO AHEAD AND TRY 40 
AND SEE WHAT HAPPENS.
WE CAN SEE 
FROM THE COMPLETED TABLE
WITH A MODIFIED DEVISOR OF 40 
EVERYTHING WORKS OUT PERFECTLY.
OKAY, LET'S GO AHEAD AND SHOW 
SOME OF THE DIVISION
TO FIND THESE QUOTAS.
THIS TIME I SHOW THE QUOTA 
FOR CHEMISTRY,
WHICH WOULD BE 135 DIVIDED 
BY 40, WHICH IS EXACTLY 3.375.
AND OF COURSE FOR BIOLOGY 
80 DIVIDED BY 40 WOULD BE 2.
SO USING THESE QUOTAS 
AND IGNORING THE DECIMALS
WE HAVE AN ALLOCATION 
OF 9, 7, 3, 2,
WHICH DOES GIVE US A SUM OF 21 
TUTORS.
THEREFORE, WE NOW HAVE THE FINAL 
ALLOCATION
USING JEFFERSON'S METHOD.
AND, AGAIN, OUR FINAL MODIFIED 
DEVISOR WAS 40.
LET'S TAKE A LOOK 
AT A SECOND EXAMPLE.
THE LEGISLATURE IN A STATE 
HAS 57 SEATS.
A PORTION OF THESE SEATS 
TO 6 COUNTIES BELOW
USING JEFFERSON'S METHOD.
FIRST STEP WE FIND THE TOTAL 
POPULATION OF ALL THE STATES
WHICH IS GIVEN HERE, 
1,000,229,000.
WE HAVE 57 STATES TO A PORTION,
SO THE STANDARD DEVISOR'S GOING 
TO BE THE TOTAL POPULATION
DIVIDED BY 57 WHICH IS GIVEN 
HERE TO THREE DECIMAL PLACES.
AND NOW TO FIND THE QUOTA 
FOR EACH COUNTY
WE'LL TAKE THE POPULATION 
OF THE COUNTY
AND DIVIDE BY OUR 
STANDARD DEVISOR,
WHICH, AGAIN, HAS ALREADY BEEN 
DONE HERE,
BUT LET'S GO AHEAD 
AND CHECK THE FIRST TWO.
SO WE HAVE 283,000 
DIVIDED BY 21,651.404,
WHICH WOULD GIVE US 
APPROXIMATELY 13.071,
WHICH WE SEE HERE.
AND THEN FOR GRANT WE WOULD HAVE 
153,000 DIVIDED BY 21,651.404,
WHICH GIVES A QUOTA OF 
APPROXIMATELY 7.067, AND SO-ON.
SO NOW FOR THE INITIAL 
APPORTIONMENT
WE IGNORE THE DECIMAL PART 
OF THE QUOTA,
SO WE HAVE 13, 7, 4, 15, 10, 
AND 5.
NOTICE HERE THE TOTAL IS 54,
BUT WE HAVE 57 SEATS 
TO A PORTION.
SO FOR THE NEXT STEP 
WE REDUCE THE STANDARD DEVISOR,
RECALCULATE THE QUOTA,
AND REPEAT THE PROCESS.
WE REPEAT THE PROCESS 
UNTIL THIS ALLOCATION HERE
SUMS TO THE TOTAL OF SEATS 
WHICH WOULD BE 57.
SO WE NEED TO REDUCE THIS.
YOU MIGHT BE ASKING 
HOW FAR TO REDUCE IT,
AND THAT'S THE CHALLENGE 
OF JEFFERSON'S METHOD.
LET'S REDUCE THE DEVISOR 
TO 20,000
AND, AGAIN, RECALCULATE 
THE QUOTAS
WHICH I'VE ALREADY DONE.
AGAIN, WE TAKE 
EACH STATE POPULATION,
DIVIDE BY 20,000 
TO GET THESE QUOTAS HERE.
AND NOW FOR THE ALLOCATION 
WE REMOVE THE DECIMAL PART,
SO WE HAVE 14, 7, 5, 17, 11, 
AND 5.
NOTICE HERE WE REDUCED 
THE DEVISOR TOO MUCH
BECAUSE NOW THE ALLOCATION 
IS FOR 59 SEATS
AND WE ONLY HAVE 57.
SO NOW WE NEED TO INCREASE 
THE DEVISOR
AND THE CHALLENGE IS HOW MUCH 
DO WE INCREASE THE DEVISOR.
LET'S SAY WE'D INCREASE 
THE DEVISOR TO 20,200,
RECALCULATE THE QUOTAS, REMOVE 
THE DECIMAL PART FOR ALLOCATION.
SO WITH A DEVISOR OF 20,200
WE HAVE AN ALLOCATION OF 14, 7, 
5, 17, 11, AND 5.
AGAIN, NOTICE 
HOW WE HAVE TOO MANY SEATS.
THIS IS 59 AND WE ONLY HAVE 57,
WHICH MEANS WE NOW INCREASE 
THE DEVISOR AGAIN.
SO LET'S TRY 20,300.
USING THE DEVISOR OF 20,300 
WE WOULD HAVE THESE QUOTAS.
AND MOVING THE DECIMAL PART WE'D 
HAVE 13, 7, 5, 17, 10, AND 5,
WHICH DOES GIVE US A SUM 
OF 57 SEATS.
SO WE'RE FINALLY DONE
AND THIS WOULD BE THE FINAL 
ALLOCATION
USING JEFFERSON'S RULE.
SO AS YOU CAN SEE, THE CHALLENGE 
ON JEFFERSON'S RULE
IS COMING UP 
WITH THE CORRECT DEVISOR
SO THAT ONCE WE CALCULATE 
THE NEW QUOTAS
AND REMOVE THE DECIMAL PART
THE ALLOCATION SUMS TO THE 
NUMBER OF SEATS AVAILABLE.
I HOPE YOU FOUND 
THIS EXPLANATION HELPFUL.
