01:640:103 Topics in Mathematics for the Liberal Arts
The Complete Exam Bible · Fall 202601:640:103 Overview
- 3 credits
- Fall 2026
- Mathematics
- SAS Core QQ and QR
- Flipped classroom format
Topics in Mathematics for the Liberal Arts is a 3-credit Rutgers course taken mainly by liberal arts and social science majors, with no prerequisite.
- Graded by Two midterms at 20% each and a 30% cumulative final, plus 30% from videos, class work and quizzes.
- Hardest step Choosing the right procedure: the same data give different answers under different methods.
- How to prepare Rework one example per chapter from a blank page and run the built-in check each method has.
- Tools A scientific calculator is needed for homework and exams; phones and algebra calculators are not allowed on exams.
How 01:640:103 is assessed
| Component | Weight | Format |
|---|---|---|
| Pre-Class Videos | 5% | Video lectures with embedded questions before each class |
| In Class Assignments | 10% | Team problem sets handed in at the end of class |
| In Class Quizzes | 15% | A quiz on the day's work at the end of each class |
| Exam 1 | 20% | Midterm listed for 10/7 in the tentative schedule |
| Exam 2 | 20% | Midterm listed for 11/11 in the tentative schedule |
| Final (cumulative) | 30% | Cumulative final, Wednesday, December 16 at 8:00 AM |
The Fall 2026 section syllabus weights the course out of 100 with the six components above. It sets grade thresholds of at least 90% for an A, 80% for at least a B, 70% for at least a C and 60% for at least a D. There are no makeups; two no-questions-asked excuses cover missed quizzes and assignments, and an excused midterm absence shifts that weight onto the matching part of the final.
Assessment structure
Segment widths reproduce the published Fall 2026 weights, which total 100%. The three exams together carry 70%.
Current 01:640:103 dates
| Date | Item | Control |
|---|---|---|
| Tuesday, September 1, 2026 | Semester begins | First day of the Fall 2026 semester. |
| Thursday, September 10, 2026 | Last day to drop without a W | As listed in the section syllabus. |
| 10/7 | Exam 1 | Class 11 in the tentative schedule. |
| 11/11 | Exam 2 | Class 21 in the tentative schedule. |
| Wednesday, December 16, 2026, 8:00 AM | Cumulative final exam | Final (cumulative), 30% of the grade. |
Dates are as published in the Fall 2026 section syllabus and its tentative schedule. Confirm exact deadlines and submission settings in the live LMS.
What 01:640:103 covers
Twelve chapters follow the Fall 2026 class schedule from voting methods to installment loans.
Preference Ballots and Voting Methods
Plurality, Instant Runoff Voting, the Borda Count and Pairwise Comparisons applied to one preference tableWhat a preference ballot records
Voters in this part of the course do not just mark a favourite. Each one ranks every candidate, with no ties, and identical ballots are grouped into the columns of a preference table.
- Preference ballots and preference tables
- Plurality and the extended ranking
- Instant Runoff Voting round by round
- The Borda Count and its points check
- Head-to-head competitions and Pairwise Comparisons
Fairness Criteria and Arrow's Impossibility Theorem
The Majority, Condorcet, Monotonicity and Independence of Irrelevant Alternatives criteria, counterexamples and Arrow's theoremCriteria judge methods; they do not elect anyone
A fairness criterion turns an intuition about who ought to win into an exact rule. The Majority Criterion says a candidate with more than half the first choices should win.
- Fairness criteria versus voting methods
- The Majority and Condorcet criteria
- Monotonicity and revotes
- Independence of irrelevant alternatives
- Arrow's impossibility theorem
Weighted Voting and the Banzhaf Power Index
Players, weights, quotas and winning coalitions; critical players; Banzhaf power; dictators, veto power and dummiesWeight is not the same as power
In a weighted voting system each player casts a number of votes, its weight, all yes or all no, and a motion passes when the yes votes reach the quota.
- Players, weights and the quota
- Coalitions and winning coalitions
- Critical players and the Banzhaf index
- Dictators, veto power and dummies
- Power from a list of winning coalitions
Shapley-Shubik Power Distribution
Sequential coalitions, pivotal players, N factorial orderings and why listing them becomes impracticalPower as the deciding vote
The Shapley-Shubik approach imagines the players joining a coalition one at a time. A sequential coalition lists every player in the order they decide to vote yes.
- Sequential coalitions
- Pivotal players
- The Shapley-Shubik power index
- Comparing Shapley-Shubik with Banzhaf
- Counting orderings with factorials
Quotas, Divisors and Hamilton's Method
Standard divisor, standard, lower and upper quotas, the Quota Rule, Hamilton's method and the Alabama paradoxThe apportionment problem
The Constitution requires House seats to be apportioned among the states according to population, but leaves the method to Congress.
- The apportionment problem
- Standard divisor and standard quotas
- Lower quotas, upper quotas and the Quota Rule
- Hamilton's method step by step
- Population per seat and the Alabama paradox
Modified Divisor Methods of Apportionment
Jefferson, Webster and Huntington-Hill: modified divisors, rounding rules, Quota Rule violations and biasNo surplus seats to hand out
The divisor methods replace Hamilton's ranking step.
- Modified divisors and why they work
- Jefferson's method
- Webster's method
- The Huntington-Hill rounding rule
- Quota Rule violations and bias
Fair Division of Continuous Goods
Fair shares by each player's own values, Divider-Chooser, Lone Divider with standoffs, and Lone ChooserFair means fair to each player
Fair division problems involve players with equal claims to goods they value differently.
- Fair shares and minimum fair shares
- Continuous division versus discrete distribution
- The Divider-Chooser method
- Lone Divider, bid lists and standoffs
- The Lone Chooser method
Fair Distribution: Sealed Bids and Markers
The Method of Sealed Bids with first and final settlements, and the Method of Markers for many small itemsIndivisible goods
A house, a car or a painting cannot be cut up without destroying its value, so each item must go whole to one player. Fairness still means that each player ends with at least one Nth of the total value by their own measure.
- Discrete goods and fair shares
- Bidding, allocation and the first settlement
- Surplus and the final settlement
- Placing markers in a row
- Allocating stretches and leftovers
Hamilton Circuits and the Traveling Salesman Problem
Weighted graphs, Hamilton circuits, (N − 1)! counting, Brute Force, Nearest Neighbor, Repetitive Nearest Neighbor and Cheapest LinkTours as weighted graphs
A trip that visits several cities and returns home can be drawn as a graph: cities are vertices, direct routes are edges, and each edge carries a weight such as a fare, a distance or a travel time.
- Graphs, weights and circuits
- Hamilton circuits and the Traveling Salesman Problem
- Counting circuits with factorials
- Brute Force and Nearest Neighbor
- Repetitive Nearest Neighbor and Cheapest Link
Trees, Networks and Kruskal's Algorithm
Connected graphs, trees, spanning trees, the V − 1 edge count, redundancy and minimum spanning trees by KruskalFrom round trips to networks
The second graph-theory question is about building a network, such as rail links between cities, so that every city can reach every other, perhaps through intermediate stops, at the least total cost.
- Connected graphs and circuits
- Trees and spanning trees
- Counting edges and redundancy
- Kruskal's Algorithm
- Why Kruskal is both optimal and efficient
Growth Factors and Compound Interest
Growth and decline factors, successive percentage changes, compound interest and FV = PV(1 + i)^nPercent changes are multiplications
Raising a quantity by 100i percent multiplies it by the growth factor 1 + i: the 1 keeps the original amount and the i adds the growth.
- Growth factors and decline factors
- Successive percentage changes
- Growth over many periods
- Compound interest versus withdrawn interest
- Present value and future value
Compounding Frequency, APY, Annuities and Loans
Periodic rates, APR versus APY, what a regular savings plan grows to, and installment loan paymentsMatch the rate to the period
Banks quote an annual percentage rate, the APR, but often compound monthly, quarterly or weekly.
- Compounding more often than yearly
- APR versus APY
- Fixed deferred annuities
- Counting payments correctly
- Installment loans and monthly payments
It studies places where mathematics is applied concretely: how elections are decided, how much power a voter really holds, how seats are apportioned, how goods are divided fairly, how to plan efficient routes and networks, and how money and populations grow.
This guide follows the Fall 2026 flipped section's class schedule chapter by chapter.
The voting and power block
The course begins with preference ballots and four voting methods, Plurality, Instant Runoff Voting, the Borda Count and Pairwise Comparisons, and then judges them with the Majority, Condorcet, Monotonicity and Independence of Irrelevant Alternatives criteria, ending with Arrow's impossibility theorem.
The power classes measure how much influence each player in a weighted voting system really has, first by the Banzhaf index and then by the Shapley-Shubik index.
Exam 1 follows these two topics in the tentative schedule.
The apportionment and fair division block
Apportionment shares indivisible seats in proportion to population: standard divisors and quotas, Hamilton's method and the Alabama paradox, then the divisor methods of Jefferson, Webster and Huntington-Hill.
Fair division gives each player a share worth at least one Nth of the goods by that player's own values, with Divider-Chooser, Lone Divider and Lone Chooser for land or time and the Methods of Sealed Bids and Markers for indivisible items.
Exam 2 follows this block.
Graphs and finance
The last classes cover Hamilton circuits and the Traveling Salesman Problem, where only Brute Force is guaranteed optimal and Nearest Neighbor and Cheapest Link trade accuracy for speed, then trees and Kruskal's Algorithm, which is both efficient and optimal.
The course closes with growth factors, compound interest, APR and APY, savings annuities and installment loans.
These two topics are examined for the first time on the cumulative final.
How to use this guide
Every chapter works complete examples with original numbers, so you can check each step with a calculator, and every method comes with a built-in check: Borda totals, power indices that sum to one, seats that sum to the house size, settlements where cash in equals cash out.
Work a chapter's example without looking, then change one number and predict what moves before recomputing.
Worked example · free
Three methods, one small election
- 2Plurality: X has 10 first choices, Z 8 and Y 5, so X wins without a majority (12 needed).
- 2Instant Runoff: Y is eliminated; its 5 ballots go to Z, which beats X 13 to 10.
- 2Borda: X 10×3 + 13×1 = 43, Y 10×2 + 8×2 + 5×3 = 51, Z 10×1 + 8×3 + 5×2 = 44; total 138 = 6 × 23.
- 2Head to head Y beats X 13 to 10 and Z 15 to 8, so Y is the Condorcet candidate.
Key terms
- Voting method
- A voting method is a rule that turns a table of preference ballots into a winner and, when extended, a full ranking.
- Majority candidate
- A majority candidate is ranked first on over half of the ballots.
- Condorcet Criterion
- The Condorcet Criterion says a candidate who beats every rival head to head should win.
- Weighted voting system
- A weighted voting system gives players different numbers of votes and passes motions that reach a quota.
- Banzhaf power index
- The Banzhaf power index measures a player's share of all critical appearances in winning coalitions.
- Standard divisor
- The standard divisor is how many people one seat stands for on average: population total over seats.
- Quota Rule
- The Quota Rule asks that every state receive its lower or its upper quota of seats.
- Fair share
- A fair share is a portion worth at least one Nth of the goods to the player who receives it.
- Minimum spanning tree
- A minimum spanning tree connects every vertex with the least possible total edge weight and no circuits.
- Annual percentage yield
- The annual percentage yield is the effective yearly growth rate once compounding within the year is counted.
01:640:103 FAQ
What topics does Topics in Mathematics for the Liberal Arts cover?
The mathematics of voting, measuring power in weighted voting, apportionment, fair division of continuous and discrete goods, Hamilton circuits and spanning trees, and growth and finance, including compound interest, annuities and loans, in that order across the semester.
How much is the final exam worth, and is it cumulative?
The final is worth 30% of the course grade and is cumulative. The Fall 2026 section syllabus lists it for Wednesday, December 16 at 8:00 AM, and the two midterms are worth 20% each.
What score do I need for an A?
The syllabus sets an A at a course score of at least 90%. At least 80% guarantees at least a B, at least 70% at least a C, and at least 60% at least a D.
Which calculator can I use on exams?
A scientific calculator is needed for homework and exams. Devices with typewriter keyboards, built-in computer algebra or wifi, including cellphones, are not permitted during exams, so practise with the calculator you will bring.
What happens if I miss a quiz or an exam?
There are no makeups, but two no-questions-asked excuses cover missed quizzes and assignments. A midterm absence excused for truly compelling reasons is not made up; instead the matching part of the final counts more heavily.
How does the flipped format work each week?
Before class you watch video lectures and answer their embedded questions. In class you work homework-level problems in a team, hand in the written work at the end, and finish with a quiz on that day's material.
What is the best way to study for this course?
Practise procedures on fresh numbers rather than rereading solutions. For each method, work one example unaided, then run its check, such as totals that must match, before moving to the next topic.
Questions students ask in each 01:640:103 chapter
- Why can different voting methods pick different winners from the same ballots?
- Each method reads a different part of the preference table. Plurality sees only the top row, Instant Runoff sees the top row repeatedly as candidates drop out, Borda weighs every position, and Pairwise Comparisons compares candidates two at a time, so a candidate strong in one reading can be weak in another. Chapter 01 · Preference Ballots and Voting Methods
- Does a Condorcet candidate always have a majority of first choices?
- No. A candidate can win every head-to-head contest while being few voters' first choice, even nobody's, if most ballots rank that candidate second. The reverse direction does hold: a majority candidate always beats each rival head to head. Chapter 02 · Fairness Criteria and Arrow's Impossibility Theorem
- Why must the quota be more than half of the total weight?
- If it were not, two groups with no members in common could each reach the quota, so contradictory motions could both pass on the same day. Setting the quota above half the total rules that out. Chapter 03 · Weighted Voting and the Banzhaf Power Index
- When do Banzhaf and Shapley-Shubik give the same distribution?
- Sometimes they coincide, as in several three-player systems, but nothing guarantees it. They measure different things, critical membership versus completing a coalition, so always state which index an answer uses. Chapter 04 · Shapley-Shubik Power Distribution
- Why can't every state simply receive its standard quota?
- Standard quotas are rarely whole numbers, and a seat cannot be split between two states. Every apportionment method is a rule for turning fractional shares into whole seats that still add to the total. Chapter 05 · Quotas, Divisors and Hamilton's Method
- How do I know which way to move the divisor?
- Compare the rounded total with the number of seats. Too few seats means the quotas are too small, so divide by a smaller number; too many means divide by a larger one. Each failed guess becomes a bound for the next. Chapter 06 · Modified Divisor Methods of Apportionment
- Why does the Divider cut the goods into exactly equal halves?
- The Divider knows nothing about the Chooser's values. Making the two halves equally valuable to themselves is the only cut that protects them whichever half the Chooser takes. Chapter 07 · Fair Division of Continuous Goods
- Do I divide by the number of items or the number of players?
- By the number of players. A minimum fair share is one Nth of a player's own total valuation, where N counts players, regardless of how many items are being distributed. Chapter 08 · Fair Distribution: Sealed Bids and Markers
- How many Hamilton circuits start at a given city when there are six cities?
- Five factorial, 5 × 4 × 3 × 2 × 1 = 120, because the other five cities can be visited in any order. Mirror images are counted separately, even though they cost the same, so a table of costs will show each total twice. Chapter 09 · Hamilton Circuits and the Traveling Salesman Problem
- How is Kruskal's Algorithm different from Cheapest Link?
- Both add edges from cheapest upward. Cheapest Link builds a circuit, so it also rejects a third edge at any vertex; Kruskal builds a tree, so it rejects only an edge that closes a circuit and allows several edges at one vertex. Chapter 10 · Trees, Networks and Kruskal's Algorithm
- Why is a 10% rise followed by a 10% fall not a return to the start?
- The fall is 10% of the higher price, so it removes more than the rise added. The factors multiply to 1.10 × 0.90 = 0.99, a net loss of 1% whichever order the two changes come in. Chapter 11 · Growth Factors and Compound Interest
- Why is the APY higher than the APR when interest compounds monthly?
- Each month's interest is added to the balance and earns interest in later months. Over a year that interest on interest lifts the total growth above the quoted annual rate. Chapter 12 · Compounding Frequency, APY, Annuities and Loans
How to study for the exam
Treat every topic as a procedure with a check. Before each exam, rework one example per chapter in that block from a blank page, name the method first, show the table the method builds, and confirm the check: Borda points, power indices summing to one, seats matching the house size, or cash in equal to cash out. For the final, add graphs and finance, which appear on no midterm.
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