text stringlengths 2 2.33k | source stringclasses 826
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ratio ofB's toA's=1ratio ofA's toB'sratio ofB's toA's=1ratio ofA's toB'sandratio ofA's toB's=1ratio ofB's toA'sratio ofA's toB's=1ratio ofB's toA's | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
units ofA=(units ofB)Ã(ratio ofA's toB's)=units ofBratio ofB's toA'sunits ofA=(units ofB)Ã(ratio ofA's toB's)=units ofBratio ofB's toA's | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
units ofB=(units ofA)Ã(ratio ofB's toA's)=units ofAratio ofA's toB'sunits ofB=(units ofA)Ã(ratio ofB's toA's)=units ofAratio ofA's toB's | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
Standard Divisor=Total PopulationHouse SizeStandard Divisor=Total PopulationHouse Size | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
State's Standard Quota=State PopulationStandard DivisorseatsState's Standard Quota=State PopulationStandard Divisorseats | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
population growth rate=current population size-previous population sizeprevious population sizepopulation growth rate=current population size-previous population sizeprevious population size | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
New House Size=New PopulationOriginal Standard Divisorrounded to the nearest whole number.New House Size=New PopulationOriginal Standard Divisorrounded to the nearest whole number. | https://openstax.org/books/contemporary-mathematics/pages/11-formula-review |
In plurality voting, the candidate with the most votes wins. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
When a voting method does not result in a winner, runoff voting can be used to do so. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on the number of candidates ranked lower than them on each ballot. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
When pairwise comparison is used, the winner will be the Condorcet candidate if one exists. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Approval voting allows voters to give equally weighted votes to multiple candidates. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic a flaw in the voting method and look for an alternative method that does not have that characteristic. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
According to Arrowâs Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not fractional, parts. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state population, total population, and house size, or the total number of seats to be apportioned. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of seats that each state should receive. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Hamiltonâs method of apportionment uses the standard divisor and standard lower quotas, and it distributes any remaining seats based on the size of the fractional parts of the standard lower quota. Hamiltonâs method satisfies the quota rule and favors neither larger nor smaller states. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Jeffersonâs method of apportionment uses a modified divisor that is adjusted so that the modified lower quotas sum to the house size. Jeffersonâs method violates the quota rule and favors larger states. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Adamsâs method of apportionment uses a modified divisor that is adjusted so that the modified upper quotas, sum to the house size. Adamsâs method violates the quota rule and favors smaller states. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Websterâs method of apportionment uses a modified divisor that is adjusted so that the modified state quotas, rounded using traditional rounding, sum to the house size. Websterâs method violates the quota rule but favors neither larger nor smaller states. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Several surprising outcomes can occur when apportioning seats that voters may find unfair: Alabama paradox, population paradox, and new-state paradox. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
Apportionment methods are susceptible to apportionment paradoxes and may violate the quota rule. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
The Balinsky-Young Impossibility Theorem indicates that no apportionment can satisfy all fairness criteria. | https://openstax.org/books/contemporary-mathematics/pages/11-key-concepts |
For the Sum of Degrees Theorem,sum of the degrees=2Ãnumber of edgessum of the degrees=2Ãnumber of edgesornumber of edges=sum of degrees2number of edges=sum of degrees2 | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
The number of edges in a complete graph withnnvertices is the sum of the whole numbers from 1 tonâ1nâ1,1+2+3+â¯+(nâ1)1+2+3+â¯+(nâ1). | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
The number of edges in a complete graph withnnvertices is1+2+3+â¯+(nâ1)=n(nâ1)21+2+3+â¯+(nâ1)=n(nâ1)2. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
The number of ways to arrangenndistinct objects isn!n!. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
The number of distinct Hamilton cycles in a complete graph withnnvertices is(nâ1)!(nâ1)!. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
In a complete graph withnnvertices, the number of distinct Hamilton cycles is(nâ1)!(nâ1)!. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
In a complete graph withnnvertices, there are at most(nâ1)!2(nâ1)!2different weights of Hamilton cycles. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
The number of edges in a tree graph withnnvertices isnâ1nâ1. A connected graph with n vertices andnâ1nâ1edges is a tree graph. | https://openstax.org/books/contemporary-mathematics/pages/12-formula-review |
Graphs and multigraphs represent objects as vertices and the relationships between the objects as edges. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The degree of a vertex is the number of edges that meet it and the degree can be zero. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
An edge must have a vertex at each end. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Multigraphs may contain loops and double edges, but simple graphs may not. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The sum of the degrees of the vertices in a graph is twice the number of edges. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
In a complete graph every pair of vertices is adjacent. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A subgraph is part of a larger graph. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Cycles are a sequence of connected vertices that begin and end at the same vertex but never visit any vertex twice. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Two graphs are isomorphic if they have the same structure. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
When graphs are relatively small, we can use visual inspection to identify an isomorphism by transforming one graph into another without breaking connections or adding new ones. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
An isomorphism between two graphs preserves adjacency. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
If two graphs differ in number of vertices, number of edges, degrees of vertices, or types of subgraphs, they cannot be isomorphic. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
When the complements of two graphs are isomorphic, so are the graphs themselves. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Walks, trails, and paths are ways to navigate through a graph using a sequence of connected vertices and edges. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Closed walks, circuits, and directed cycles are ways to navigate from a vertex on a graph and return to the same vertex. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Colorings are a way to organize the vertices of a graph into groups so that no two members of a group are adjacent. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Maps can be represented with planar graphs, which can always be colored using four colors or fewer. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A connected graph has only one component. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The Euler circuit theorem states that an Euler circuit exists in every connected graph in which all vertices have even degree, but not in disconnected graphs or any graph with one or more vertices of odd degree. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The Chinese postman problem asks how to find the shortest closed trail that visits all edges at least once. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
If an Euler circuit exists, it is always the best solution to the Chinese postman problem. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Eulerization is the process of adding duplicate edges to a graph so that the new multigraph has an Euler circuit. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The minimum number of duplicated edges needed to eulerize a graph is half the number of odd vertices or more. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
An Euler trail exists whenever a graph has exactly two vertices of odd degree. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
When a bridge is removed from a graph, the number of components increases. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A bridge is never part of a circuit. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
When a local bridge is removed from a graph, the distance between vertices increases. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
An edge that is part of a triangle is never a local bridge. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A Hamilton cycle is a directed cycle, or circuit, that visits each vertex exactly once. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Some Hamilton cycles are also Euler circuits, but some are not. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Hamilton cycles that follow the same undirected cycle in the same direction are considered the same cycle even if they begin at a different vertex. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The number of unique Hamilton cycles in a complete graph with n vertices is the same as the number of ways to arrangenâ1nâ1distinct objects. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Weighted graphs have a value assigned to each edge, which can represent distance, time, money and other quantities. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A Hamilton path visits every vertex exactly once. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Some Hamilton paths are also Euler trails, but some are not. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient but usually does not lead to the ideal solution. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The brute force method finds a Hamilton cycle of lowest weight in a complete graph. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a complete graph. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A brute force algorithm always finds the ideal solution but can be impractical whereas a greedy algorithm is efficient but usually does not lead to the ideal solution. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
A Hamilton cycle of lowest weight is a solution to the traveling salesperson problem. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The brute force method finds a Hamilton cycle of lowest weight in a complete graph. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
The nearest neighbor method is a greedy algorithm that finds a Hamilton cycle of relatively low weight in a complete graph. | https://openstax.org/books/contemporary-mathematics/pages/12-key-concepts |
Place Valueas inFigure 1.3. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Name a Whole Number in WordsStep 1.Start at the left and name the number in each period, followed by the period name.Step 2.Put commas in the number to separate the periods.Step 3.Do not name the ones period. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Start at the left and name the number in each period, followed by the period name. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Put commas in the number to separate the periods. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Do not name the ones period. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Write a Whole Number Using DigitsStep 1.Identify the words that indicate periods. (Remember the ones period is never named.)Step 2.Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.Step 3.Name the number in each period and place the digits in the correct place value po... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Identify the words that indicate periods. (Remember the ones period is never named.) | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Name the number in each period and place the digits in the correct place value position. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Round Whole NumbersStep 1.Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.Step 2.Underline the digit to the right of the given place value.Step 3.Is this digit greater than or equal to 5?Yesâadd 1 to the digit in the given place value.Noâdonotchange the digi... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 1.Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 2.Underline the digit to the right of the given place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 3.Is this digit greater than or equal to 5?Yesâadd 1 to the digit in the given place value.Noâdonotchange the digit in the given place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Yesâadd 1 to the digit in the given place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Noâdonotchange the digit in the given place value. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Step 4.Replace all digits to the right of the given place value with zeros. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Divisibility Tests:A number is divisible by:2 if the last digit is 0, 2, 4, 6, or 8.3 if the sum of the digits is divisible by 3.5 if the last digit is 5 or 0.6 if it is divisible by both 2 and 3.10 if it ends with 0. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
2 if the last digit is 0, 2, 4, 6, or 8. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
3 if the sum of the digits is divisible by 3. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
5 if the last digit is 5 or 0. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
6 if it is divisible by both 2 and 3. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
10 if it ends with 0. | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
Find the Prime Factorization of a Composite NumberStep 1.Find two factors whose product is the given number, and use these numbers to create two branches.Step 2.If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.Step 3.If a factor is not prime, write it as the product of two factors... | https://openstax.org/books/elementary-algebra-2e/pages/1-key-concepts |
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