Content code
m1523
Slug (identifier)
memory-aid-mathematics-secondary-5-cst
Grades
Secondary V
Topic
Mathematics
Content
Contenu
Corps

The following is a short preparation guide containing all the concepts covered in Secondary 5 in the CST pathway. To explain everything, each formula will be followed by an example and a link to a concept sheet in our virtual library.

Columns number
3 columns
Format
33% / 33% / 33%
First column
Title
Algebra
Links
Title
Financial Mathematics
Links
Second column
Title
Probability
Links
Title
Statistics: Social Choice Theory
Links
Third column
Title
Geometry
Links
Title
Graphs
Links
Title (level 2)
Algebra
Title slug (identifier)
algebra
Contenu
Title (level 3)
First Degree Linear Function (Functional and General Form)
Title slug (identifier)
first-degree-linear-function
Content
Columns number
2 columns
Format
50% / 50%
First column
Corps

Functional form

||y = ax + b|| where ||a = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}||

Second column
Corps

General form

|0 = Ax + By + C| where |A, B, C \in \mathbb{Z}|

Content
Corps

Using the information in the table below, determine the equation for the straight line in the general form.

Image
Image.
Solution
Corps
  1. Find the slope according to |\dfrac{\Delta y}{\Delta x}​.|
    ||a = \dfrac{2 - 4{.}4}{-0{.}5 - 0{.}5} = 2{.}4||

  2. Find the initial value |(b)| by substituting in one of the points on the graph.
    ||​\begin{align} f(x) &= 2{.}4 x + b\\ 2 &= 2{.}4 (-0{.}5) + b \\ b &= 3{.}2 \end{align}||

  3. Transform the value of parameters |a| and |b| to their simplified fractional form.
    ||​\begin{align} y &= 2{.}4x + 3{.}2 \\ y &= \dfrac{24}{10}x + \dfrac{32}{10} \\ y &= \dfrac{12}{5}x + \dfrac{16}{5} \end{align}||

  4. Find a common denominator for all the terms in the equation.
    ||​\begin{align} y &= \dfrac{12}{5}x + \dfrac{16}{5} \\ \Rightarrow\ \frac{5y}{5} &= \dfrac{12x}{5} + \dfrac{16}{5} \end{align}||

  5. Make the equation equal to |0.|
    ||​\begin{align} \frac{5y}{\cancel{5}} &= \frac{12x}{\cancel{5}} + \frac{16}{\cancel{5}} \\ \Rightarrow 5y &= 12x + 16 \\ \Rightarrow\, \ 0 &= 12x - 5y + 16 \end{align}||

Answer: The equation of the straight line in general form is |0 = 12x - 5y + 16.|

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See also

Links
Title (level 3)
Solving a System of Equations (Comparison)
Title slug (identifier)
solving-a-system-of-equations-comparison
Content
Corps

At the corner store, workers bought 4 coffees and 6 muffins for $15.06. The next day, the same group buys 3 coffees and 5 muffins for $11.97. If, the next day, these workers want to buy 6 coffees and 4 muffins, how much will they have to pay?

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See also

Links
Title (level 3)
Solving a System of Equations (Substitution)
Title slug (identifier)
solving-a-system-of-equations-substitution
Content
Corps

Take the following steps to solve a system of equations by substitution:
1) Identify the variables as they relate to the unknowns
2) Create the equations according to the scenario
3) Isolate a variable in one of the two equations
4) Substitute the same variable in the other equation by the algebraic expression associated with it
5) Solve the new equation
6) Replace the value of the variable found in step 5 in one of the starting equations to find the value of the other variable

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See also

Links
Title (level 3)
Solving a System of Equations (Elimination)
Title slug (identifier)
solving-a-system-of-equations-elimination
Content
Corps

Follow these steps to solve an equation system by elimination:
1) Identify the variables as they relate to the unknowns
2) Create the equations according to the scenario
3) Find equivalent equations to obtain the same coefficient of the same variable
4) Subtract the two equations
5) Isolate the remaining variable to find its value
6) Replace the value of the variable found in Step 5 into one of the starting equations to find the value of the other variable

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See also

Links
Title (level 3)
Optimization
Title slug (identifier)
optimization
Corps

 

 

Content
Corps

Generally, an optimization problem can be solved by following these steps:

  1. Identify variables
     
  2. Create the system of inequalities
     
  3. Determine the equation of the function to be optimized
     
  4. Draw the polygon of constraints
     
  5. Determine the coordinates of each of the polygon’s vertices
     
  6. Evaluate the function to be optimized for each vertex of the polygon of constraints
     
  7. Determine the coordinate(s) of the point(s) that optimize(s) the function
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See also

Links
Title (level 2)
Probability
Title slug (identifier)
probability
Contenu
Corps
TYPES OF EVENTS ​DEFINITION ​EXAMPLE
Mutually exclusive ​When they can't happen at the same time. Rolling a six-sided die and getting a number that is a multiple of both 3 and 4. 
​Not mutually exclusive When they can occur at the same time. ​Drawing a random card from a 52-card deck and getting an ace that is red.
Dependent When one result affects the other.  ​Drawing two cards successively without replacing it from a 52-card deck.
Independant When one result does not affect the other. ​Drawing a card from a 52-card deck and rolling a six-sided die.
Content
Corps

However, we must not forget the types of events that have been seen in class in previous years (certain, probable, impossible, elementary, complementary, compatible, and incompatible).

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See also

Links
Title (level 3)
The Odds For and the Odds Against
Title slug (identifier)
odds-for-and-odds-against
Content
Corps

If |a = | the chances for and |b =| the odds against, then:

 
- the odds (ratio) for |\displaystyle = a : b \Rightarrow \frac {a}{a+b}|
 
- the odds (ratio) against |\displaystyle = b : a \Rightarrow \frac {b}{b+a}| 
 

Thus, we obtain the net winnings according to the following proportion:

|\displaystyle \frac{\text{Bet amount}}{\text{Net gain}} = \frac{\text{Number of chances on which we bet}}{\text{Total number of chances}}| 
Content
Title (level 3)
Example of Chances For
Title slug (identifier)
example-of-chances-for
Corps

In the days of the Quebec horse races, people could bet on the victories of racehorses. Each horse had odds that quantified its chances of winning. For the last race, an amateur bet |$20| for victory whose odds were |1:14.| So, what was the potential gain from his bet?

Solution
Corps
  1. ​Apply the proportion ||​\begin{align} \dfrac{20}{\text{Gain net}} &= \dfrac{\color{blue}{1}}{\color{blue}{1}+\color{red}{14}}\\\\ \dfrac{20}{\text{Gain net}} &= \dfrac{\color{blue}{1}}{15}\end{align}||

  2. Solve with the cross product ||​\begin{align} \dfrac{20}{\text{Gain net}} &= \dfrac{\color{blue}{1}}{15}\\\\ \Rightarrow\ \text{Gain net} &= \frac{20 \times 15}{\color{blue}{1}} \\ \text{Gain net} &= 300\end{align}||

​​Answer: If his horse finished first in the race, the amateur would walk away with the sum of |$300.|

Content
Title (level 3)
Example of Against Chances
Title slug (identifier)
example-of-against-chances
Corps

In some boxing matches, you can bet on a boxer's defeat. Each boxer has odds that quantify his chances of winning. For the next fight, the champion has odds of |44 : 1| for his victory. So what would be the net profit of an amateur who bets |$10| against a victory for the champion?

Solution
Corps
  1. ​Identifying the odds against ||​\begin{align} \text{Odds ratio} &= \color{blue}{44} : \color{red}{1} \\ \Rightarrow\ \text{Odds ratio} &= \color{red}{1} : \color{blue}{44}\end{align}||

  2. Apply the proportion ||​\begin{align} \dfrac{10}{\text{Net gain}} &= \dfrac{\color{red}{1}}{\color{red}{1}+\color{blue}{44}} \\\\ \dfrac{10}{\text{Net gain}} &= \dfrac{\color{red}{1}}{45}\end{align}||

  3. Solve with the cross product ||​\begin{align} \dfrac{10}{\text{Net gain}} &= \dfrac{\color{red}{1}}{45} \\\\ \Rightarrow \text{Net gain} &= \frac{10 \times 45}{\color{red}{1}} \\ \text{Net gain} &= 450\end{align}||

Answer: If the champion fails to keep his belt, the boxing fan wins |$450.|

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See also

Links
Title (level 3)
Mathematical Expectation (Fair Games)
Title slug (identifier)
mathematical-expectation
Content
Corps
|\mathbb{E} = (p_1 x_1 + p_2 x_2 + \dots + p_i x_i) - M|

where |p_i =| probability of occurrence of the event |i|,  |x_i =| amount associated with the event |i|, |M =| initial bet amount

Content
Corps

To help fund the school's freestyle ski team, the organisers are organising a fundraising event with the following prizes up for grabs.

  • one weekend family ski package ($800 value);

  • two alpine ski season tickets (value of $500 each);

  • four pairs of skis (value of $300 each);

  • eight lift tickets valid for one day (value $45 each).

Given that they have a total of 336 tickets to sell, what should be the selling price of a raffle ticket?

Solution
Corps
  1. Apply the mathematical expectation formula

|\begin{align}\mathbb{E} &= (\color{blue}{p_1 x_1} + \color{red}{p_2 x_2} + \color{green}{p_3 x_3} + \color{black}{p_4 x_4}) - M \\\\ \displaystyle \mathbb{E} &= \left(\color{blue}{\frac{1}{336}\times 800 } + \color{red}{\frac{2}{336} \times 500} + \color{green}{\frac{4}{336} \times 300} + \color{black}{ \frac{8}{336} \times 45}\right) - M\end{align}|​

  1. ​Replace the value of |\mathbb{E}| by |0| because the game is fair

​|\begin{align}\displaystyle \mathbb{E} &= \left(\color{blue}{\frac{1}{336}\times 800 } + \color{red}{\frac{2}{336} \times 500} + \color{green}{\frac{4}{336} \times 300} + \color{black}{ \frac{8}{336} \times 45}\right) - M \\\\ \displaystyle 0​ &= \left(\color{blue}{\frac{1}{336}\times 800 } + \color{red}{\frac{2}{336} \times 500} + \color{green}{\frac{4}{336} \times 300} + \color{black}{ \frac{8}{336} \times 45}\right) - M\end{align}|

  1. ​Isolate |M| to find the value of the initial stake

|\begin{align}\displaystyle 0​ &= \left(\color{blue}{\frac{1}{336}\times 800 } + \color{red}{\frac{2}{336} \times 500} + \color{green}{\frac{4}{336} \times 300} + \color{black}{ \frac{8}{336} \times 45}\right) - M\\\\ \displaystyle 0​ &= \left(\color{blue}{\frac{800}{336}} + \color{red}{\frac{1000}{336}} + \color{green}{\frac{1200}{336}} + \color{black}{ \frac{360}{336}}\right) - M\\\\ \displaystyle 0 &= \frac{3360}{336} - M\\\\ \displaystyle M &= \frac{3360}{336} \\\\ M &= 10\ $\end{align}|

Answer: For the game to be fair, tickets must be sold at a price of |$10.|

Content
Corps

If |\mathbb{E} = 0|, the game is fair.
If |\mathbb{E} < 0|, the game is unfavourable to the player.
If |\mathbb{E} >0|, the game is favourable to the player.

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See also

Links
Title (level 3)
Conditional Probability
Title slug (identifier)
conditional-probability
Content
Corps

|P(A \mid B) = \dfrac{P (A \cap B)}{P (B)}| with |P(A) >0|​

Content
Corps

During the previous month, listeners to a Quebec radio station had the chance to win a trip to Walt Disney's fairytale estate. Before randomly selecting the winner, the broadcaster drew up an overall profile of the participants.

m1522i16.PNG

English image coming soon!

What is the probability that the winner is the father of a family with three children and that the raffle ticket was given to him as a gift?

Solution
Corps
  1. Identify the boxes referring to the participants who received the ticket as a gift
    m1522i17.PNG

​||P(\color{red}{B}) = \dfrac{\color{red}{15 + 30 + 2}}{23 + 12 + ... + 67 + 27  } = \dfrac{\color{red}{47}}{240}||

  1. Among the people identified above, identify those with a family of three children
    m1522i18.PNG

||P(\color{blue}{A \cap B}) = \dfrac{\color{blue}{30}}{240}||

  1. Apply the formula

||P(\color{blue}{A} \mid \color{red}{B}) = \frac {P( \color{blue}{A \cap B})}{P(\color{red}{B})}= \dfrac{\frac{\color{blue}{30}}{240}}{\frac{\color{red}{47}}{240}}= \dfrac{\color{blue}{30}}{\color{red}{47}}||

Answer: The probability that the winner is the father of a family with three children and that he was given the ticket as a gift is |\dfrac{30}{47}.|

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See also

Links
Title (level 2)
Geometry
Title slug (identifier)
geometry
Contenu
Title (level 3)
Equivalent Figures
Title slug (identifier)
equivalent-figures
Content
Corps

Two figures are equivalent when they have the same area.

Text

See also

Links
Title (level 3)
Equivalent Solids
Title slug (identifier)
equivalent-solids
Content
Corps

Two solids are equivalent when they have the same volume.

Text

See also

Links
Title (level 2)
Financial Mathematics
Title slug (identifier)
financial-mathematics
Contenu
Title (level 3)
Properties of Exponential Notation
Title slug (identifier)
properties-exponential-notation
Content
Corps

Solve the following equation. ||6 \ 300 (1{.}2)^{3x} = 175 (7{.}2)^2||

Solution
Corps
Calculs Explanations

|\begin{align}
6 \ 300 (1{.}2)^{3x} &= 175 (7{,}2)^2 \\
&= 175 (6 \times 1{.}2)^2 \end{align}|

Factorisation to find equivalent bases

|\begin{align}
\phantom{6 \ 300 (1{.}2)^{3x}}&= 175 (6 \times1{.}2)^2 \\
&= 175 \times(6)^2 \times(1{.}2)^2 \end{align}|

Power of a product

|\begin{align}
\phantom{6 \ 300 (1{.}2)^{3x}} &= 175 \times (6)^2 \times (1{.}2)^2 \\
&=6 \ 300 (1{.}2)^2
\end{align}|

Power calculation and multiplication

|\begin{align}
\dfrac{6 \ 300 (1{.}2)^{3x}}{\color{red}{6 \ 300}} &=\dfrac{6 \ 300 (1{.}2)^2}{\color{red}{6 \ 300}}\\1{.}2^{3x} &= 1{.}2^2\end{align}|

Reciprocal operation to isolate exponential notation

|\begin{align}\phantom{(6 \ 300)}1{.}2^{\color{blue}{3x}} &= 1{.}2^\color{blue}{2} \\ \dfrac{\color{blue}{3x}}{\color{red}{3}} &= \dfrac{\color{blue}{2}}{\color{red}{3}} \\ x &= \dfrac{2}{3} \end{align}|

Comparison of exponents with identical bases

Answer : |x=\dfrac{2}{3}.|

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See also

Links
Title (level 3)
Definitions and Laws of Logarithmic Notation
Title slug (identifier)
definitions-laws
Content
Corps

Solving an exponential equation 
What is the value of |x| in the equation:
|| 4500 = 1500 (1.08)^{\frac{x}{2}}||
||\begin{align}
\frac{4500}{\color{red}{1500}} &= \frac{1500}{\color{red}{1500}} (1.08)^{\frac{x}{2}} && \small\text{inverse operation} \\\\
3 &= 1.08^{\frac{x}{2}} \\\\
log_{1.08} \ 3 &= \frac{x}{2} && \small \text{definition of log} \\\\
\frac{log_{10} \ 3}{log_{10} \ 1.08} &= \frac{x}{2} && \small\text{change of base} \\\\
14.275 \cdot \color{red}{2} &\approx \frac{x}{2} \cdot \color{red}{2} && \small\text{inverse operation} \\\\
28.55 &\approx x \end{align}||

Solving a logarithmic equation 
What is the value of |x| in the equation:
||log_5 \ x^3 + log_5 \ \left(\frac{x}{32}\right) = log_5 \ 732 - 1||
||\begin{align}
log_5 \ x^3 + log_5 \ \left(\frac{x}{32}\right) &= log_5 \ 732 - 1 \\\\
3 \ log_5 \ x + log_5 \ \left(\frac{x}{32}\right) &= log_5 \ 732 - 1 && \small\text{power of a log}\\\\
 3 \ log_5 \ + (log_5 \ x - log_5 \ 32) &= log_5 \ 732 - 1 && \small\text{log of a quotient} \\\\
3 \ log_5 \ x + log _5 \ x - 2,153 &\approx 4.098 - 1 && \small\text{change of base law} \\\\
4 \ log_ 5 \ x - 2.153 &\approx 4.098 - 1 && \small\text{like terms} \\\\
4 \ log_5 \ x - 2.153 \color{red}{+2.153} &\approx 4,098 - 1 \color{red}{+ 2.153} && \small\text{inverse operation} \\\\
\frac{4 \ log_5 \ x}{\color{red}{4}} &\approx \frac{5.251}{\color{red}{4}} && \small\text{inverse operation} \\\\
log_5 \ x \approx 1.313 &\Rightarrow 5^{1.313} = x && \small\text{definition of log} \\\\
8.275 &\approx x \end{align}||
 

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Links
Title (level 3)
Finding Current Value, Calculating Future Value and Modeling a Financial Situation
Title slug (identifier)
finding-current-value-calculating-future-value-and-modeling-a-financial-situation
Corps

Finding Current Value

Content
Corps

||\begin{align}
C_0&=C_n\left(1+\dfrac{i}{k}\right)^{-x}\\\\
C_0&=\dfrac{C_n}{\left(1+\dfrac{i}{k}\right)^x}\\\\
\end{align}||

where

||\begin{align}
C_0&:\text{present value}\\
C_n&:\text{future value}\\
i&:\text{annual interest rate in decimal notation}\\
k&:\text{factor related to the interest period}\\
x&:\text{number of interest periods}
\end{align}||

Content
Corps

To ensure the best possible retirement, Christian needs to obtain a future value of |$200\ 000| on an investment he is making today. So what should be the present value of his investment if he knows that it will be subject to an annual interest rate of |2{.}59\ \%| compounded monthly over a period of |35| years?

Solution
Corps
  1. Identify the differents types of data ||\begin{align} \color{#ec0000}{C_n}&=\color{#ec0000}{200\ 000}\\ \color{#333fb1}{i}&=\color{#333fb1}{2{,}59\ \%}\\ \color{#fa7921}{k}&=\color{#fa7921}{12}\\ \color{#ff55cc}{x}&=\color{#ff55cc}{35\times12}=\color{#ff55cc}{420} \end{align}||

  2. Apply the formula ||\begin{align} C_0&=\dfrac{\color{#ec0000}{C_n}}{\left(1+\dfrac{\color{#333fb1}{i}}{\color{#fa7921}{k}}\right)^{\color{#ff55c3}{x}}}\\\\ C_0&=\dfrac{\color{#ec0000}{200\ 000}}{\left(1+\dfrac{\color{#333fb1}{0{.}0259}}{\color{#fa7921}{12}}\right)^{\color{#ff55c3}{420}}}\\\\ C_0&\approx80\ 866{.}06 \end{align}||

  3. Give the answer in a sentence

    The present value of Christian's investment should be approximately |$80\ 866{.}06.|

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See also

Links
Title (level 2)
Statistics: Social Choice Theory
Title slug (identifier)
statistics
Contenu
Title (level 3)
Majority Rule
Title slug (identifier)
majority-rule
Corps

This procedure confers victory on the individual or group that obtains the majority of votes, i.e. |50\ \%+1| of the total number of votes representing an absolute majority.

Content
Corps

In the last Canadian Federal Election, the aim of the various parties was to elect as many deputies as possible from the country's 338 ridings. Once the results have been compiled, here's how power is distributed.

Political Party

Number of Elected Deputies

Conservative Party

|125|

Green Party

|4|

Liberal Party

|171|

New Democratic Party

|16|

Bloc Québécois Party

|22|

Solution
Corps
  1. Determine the percentage of each party using the following proportion ||\dfrac{\text{Number of elected deputies}}{\text{Number of elected deputies}}=\dfrac{\text{% of Party}}{\text{100 %}}||

    Political Party

    Percentage of Elected Deputies (%)

    Conservative Party

    |37{.}0|

    Green Party

    |1{.}2|

    Liberal Party

    |50{.}6|

    New Democratic Party

    |4{.}7|

    Bloc Québécois Party

    |6{.}5|

  2. Identify the group or individual who received more than 50% of the vote

    Political Party

    Percentage of Elected Deputies (%)

    Conservative Party

    |37{.}0|

    Green Party

    |1{.}2|

    Liberal Party

    |50{.}6|

    New Democratic Party

    |4{.}7|

    Bloc Québécois Party

    |6{.}5|

Under majority rule, the Liberal Party of Canada won the election.

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See also

Links
Title (level 3)
The plurality rule
Title slug (identifier)
the-plurality-rule
Corps

This procedure confers victory on the individual or group that obtains the greatest number of votes, i.e. an absolute majority.

Content
Corps

To elect the new captain of the Montreal Canadiens hockey team, the general manager asked the opinion of all the players who have a contract with the team. Each player was asked to write down the name of the player they wanted to lead the team. Here are the results compiled by management.

Player

Number of Votes

Brendan Gallagher

2

P.K. Subban

1

Max Pacioretty

16

David Desharnais

5

Tomas Plekanec

1

Andrei Markov

11

Using the plurality method, who will be named captain of this team?

Solution
Corps
  1. Place results in descending order

Player

Number of Votes

Max Pacioretty

16

Andrei Markov

11

David Desharnais

5

Brendan Gallagher

2

P.K. Subban

1

Tomas Plekanec

1

The team captain will be Max Pacioretty, who received the most votes.

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See also

Links
Title (level 3)
The Borda method
Title slug (identifier)
borda
Corps

This procedure confers victory on the individual or group who obtains the most points by awarding| n-1| points for each voter's 1st choice, |n-2| points for the 2nd choice and so on for the next choices |n| of candidates.

Content
Corps

To be inducted into the Hockey Hall of Fame, the players nominated are ranked in order of preference by the members of the Hall committee. To simplify the presentation of the results, votes with identical preferences have been grouped together. Here is the list of four players and their preference ranking in 2014.

Choice

7 Members

6 Members

4 Members

1st choice

Dominik

Mike

Peter

2nd choice

Peter

Peter

Rob

3rd choice

Mike

Dominik

Mike

4th choice

Rob

Rob

Dominik

Using Borda's method, who would be the next player to be inducted into the Hockey Hall of Fame?

Solution
Corps
  1. Determine the number of points awarded for each choice

    We use the following rule. ||\text{Nombre de points}=n-p||

    where
    |n :| number of possible choices
    |p :| position on the preference list

    1st : |n-p=4-1=3| points
    2nd : |n-p=4-2=2| points
    3rd : |n-p=4-3=1| point
    4th : |n-p=4-4=0| point

  2. Calculate the number of points obtained by each

    Dominik :|\color{#333fb1}{7}\times(4-1)+\color{#ec0000}{6}\times(4-3)+\color{#3a9a38}{4}\times(4-4)=27\text{ points}|
    Peter : |\color{#333fb1}{7}\times(4-2)+\color{#ec0000}{6}\times(4-2)+\color{#3a9a38}{4}\times(4-1)=38\text{ points}|
    Mike : |\color{#333fb1}{7}\times(4-3)+\color{#ec0000}{6}\times(4-1)+\color{#3a9a38}{4}\times(4-3)=29\text{ points}|
    Rob : |\color{#333fb1}{7}\times(4-4)+\color{#ec0000}{6}\times(4-4)+\color{#3a9a38}{4}\times(4-2)=8\text{ points}|

Since Peter scored the most points, he is the only player who will be eligible for the Hockey Hall of Fame in 2014.

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See also

Links
Title (level 3)
The Condorcet's Principle
Title slug (identifier)
the-condorcet-principle
Corps

This procedure confers victory on the individual or group that wins all its head-to-head duels according to voters' preferences.

If no individual or group wins all their duels, it is preferable to use another procedure.

Content
Corps

To be inducted into the Hockey Hall of Fame, the players nominated are ranked in order of preference by the members of the Hall committee. To simplify the presentation of the results, votes with identical preferences have been grouped together. Here is the list of four players and their preference ranking in 2014.

Choice

7 Members

6 Members

4 Members

1st choice

Dominik

Mike

Peter

2nd choice

Peter

Peter

Rob

3rd choice

Mike

Dominik

Mike

4th choice

Rob

Rob

Dominik

Using Condorcet's principle, who would be the next player to be inducted into the Hockey Hall of Fame?

Solution
Corps
  1. Compile Dominik's duels with each of the players

    Dominik |(\color{#333fb1}{7})| is beaten by Peter |(\color{#ec0000}{6}+\color{#3a9a38}{4}).|

    Since he loses his first duel, we can move on to another candidate.

  2. Compile Peter's duels with each player

    Peter |(\color{#ec0000}{6}+\color{#3a9a38}{4})| wins against Dominik |(\color{#333fb1}{7}).|

    Peter |(\color{#333fb1}{7}+\color{#3a9a38}{4})| wins against Mike |(\color{#ec0000}{6}).|

    Peter |(\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4})| wins against Rob |(0).|

Since Peter won all his duels, he will be inducted into the Hockey Hall of Fame.

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See also

Links
Title (level 3)
Assent Voting
Title slug (identifier)
assent-voting
Corps

This procedure confers victory on the individual or group that obtains the highest number of votes while voters can cast their ballot only once for as many candidates as they wish.

Content
Corps

In order to avoid the popular vote, the assent vote is used to elect the next class president. Once the results have been counted, the following table is obtained.

Number of Voters who Voted for these Candidates

5

8

10

7

3

 

Marie-Claude

Simon

Vincent

Judith

Simon

 

Gitane

Vincent

Gitane

Marie-Claude

Judith

 

 

Gitane

Simon

Vincent

 

If we compiled the results properly, who would be the winner of this election if we followed the principle of assent-based voting?

Solution
Corps
  1. Compiling Gitane's votes ||\color{#ec0000}{5}+\color{#3a9a38}{8}+\color{#ff55c3}{10}=23\text{ votes}||

  2. Compiling Marie-Claude's votes ||\color{#ec0000}{5}+\color{#333fb1}{7}=12\text{ votes}||

  3. Compiling Simon's votes ||\color{#3a9a38}{8}+\color{#efc807}{3}+\color{#ff55c3}{10}=21\text{ votes}||

  4. Compiling Vincent's votes ||\color{#3a9a38}{8}+\color{#333fb1}{7}+\color{#ff55c3}{10}=25\text{ votes}||

  5. Compiling Judith's votes ||\color{#333fb1}{7}+\color{#efc807}{3}=10\text{ votes}||

Since Vincent received the most votes (|25| votes), he will be appointed class president.

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See also

Links
Title (level 3)
Voting by Elimination
Title slug (identifier)
voting-by-elimination
Corps

This procedure confers victory on the individual or group that obtains the majority of votes, i.e. |50\ \%+1| of the total number of votes, while voters rank them in order of preference. If there is no absolute majority on the first count, we eliminate the least popular to transfer votes to the next candidate.

Content
Corps

To be inducted into the Hockey Hall of Fame, the players nominated are ranked in order of preference by the members of the Hall committee. To simplify the presentation of the results, votes with identical preferences have been grouped together. Here is the list of four players and their preference ranking in 2014.

Choice

7 Members

6 Members

4 Members

1st choice

Dominik

Mike

Peter

2nd choice

Peter

Peter

Dominik

3rd choice

Mike

Dominik

Mike

Using the elimination voting method, who would be the next player to be inducted into the Hockey Hall of Fame?

Solution
Corps
  1. Determine whether one of the candidates has the majority of 1st choices

    Dominik ||\dfrac{\color{#333fb1}{7}}{\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4}}=\dfrac{\color{#333fb1}{7}}{17}\approx41{.}2\ \%||

    Mike ||\dfrac{\color{#ec0000}{6}}{\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4}}=\dfrac{\color{#ec0000}{6}}{17}\approx35{.}3\ \%||

    Peter ||\dfrac{\color{#3a9a38}{4}}{\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4}}=\dfrac{\color{#3a9a38}{4}}{17}\approx23{.}5\ \%||

  2. Since no one has a majority of votes, the votes of the least popular are transferred to the next most popular member of the group

    Choice

    7 Members

    6 Members

    4 Members

    1st choice

    Dominik

    Mike

    Peter

    2nd choice

    Peter

    Peter

    Dominik

    3rd choice

    Mike

    Dominik

    Mike

  3. Redo the calculations to determine whether one of the remaining candidates has a majority of votes

    Dominik ||\dfrac{\color{#333fb1}{7}+\color{#3a9a38}{4}}{\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4}}=\dfrac{11}{17}\approx64{.}7\ \%||

    Mike ||\dfrac{\color{#ec0000}{6}}{\color{#333fb1}{7}+\color{#ec0000}{6}+\color{#3a9a38}{4}}=\dfrac{\color{#ec0000}{6}}{17}\approx35{.}3\ \%||

Since Dominik has now obtained more than |50\ \%| of votes, he will be inducted into the Hockey Hall of Fame.

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See also

Links
Title (level 2)
Graphs
Title slug (identifier)
graphs
Contenu
Content
Corps

To fully understand the concepts in this section, it is important to master the following vocabulary.

  • Vertices : the different elements which are linked (people, steps to be taken, etc.) and which are generally represented by dots.

  • Edges : the links that connect the elements, generally represented by lines or arcs.

  • Parallel edges : when two edges have the same start and end vertices.

  • Loop : edge which begins and ends with the same vertex.

  • Degree : the number of times a vertex is touched by the different edges.

  • Chain : a sequence of edges that we loan to ‘walk’ around the graph.

  • Length : corresponds to the number of edges loaned in a chain.

  • Distance​ : minimum number of stops to go from the starting vertex to the destination vertex.

  • Simple chain : a chain in which each edge is loaned only once.

  • Cycle ​: a chain that starts and ends at the same vertex.

  • Simple cycle : a cycle in which each edge is used only once.

Content
Columns number
2 columns
Format
50% / 50%
First column
Image
A graph containing parallel edges and a loop.
Second column
Corps

|\color{#ec0000}{B}| is a vertex.

|\color{#333fb1}{A — E}| is an edge.

|\color{#3a9a38}{F — F}| is a loop.

|E — D| and |D — E| are parallel edges.

The degree of |\color{#ec0000}{B}| is |3.|

Columns number
2 columns
Format
50% / 50%
First column
Image
A graph containing a cycle and a chain
Second column
Corps

|\color{#ec0000}{B-F-E-C-F-B}| is a cycle.

|\color{#3a9a38}{D-B-C-B-A}| is a chain.

Columns number
2 columns
Format
50% / 50%
First column
Image
A graph containing a cycle
Second column
Corps

|\color{#3a9a38}{A-B-C-F}| is a simple string of length |3,| but the distance |\color{#333fb1}{d(A,F)=2}.|

Finally, |\color{#3a9a38}{A-B-C-F}\color{#333fb1}{-E-A}| is a simple cycle of length |5.|

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See also

Links
Title (level 3)
The Chain and the Eulerian Cycle
Title slug (identifier)
the-chain-and-the-eulerian-cycle
Corps

The Eulerian characteristic of a graph requires that all edges are involved only once in the chain or cycle.

Content
Corps

As a police officer, you want to know every nook and cranny of the area you serve. To do this, you decide to patrol every street in your district during your shift. To help you, you use a road map to identify the area you have to supervise.

Image
 A graph of a police officer's route.
Corps

Bearing in mind that you can decide where your route starts and finishes, what sequence of routes should you take to patrol each street as efficiently as possible?

Solution
Corps
  1. Choose a starting point that seems appropriate.

Image
A graph with an identified starting point.
Corps
  1. Make sure that all the edges are loaned only once. However, it is possible to pass through the same vertex several times.

Image
A graph containing an Eulerian chain.
Corps

One possible route could be the following Eulerian chain. ||\color{#ec0000}{A}-B-F-\color{#ec0000}{A}-E-C-B-D-E||

Text

See also

Links
Title (level 3)
The Chain and the Hamiltonian Cycle
Title slug (identifier)
the-chain-and-the-hamiltonian-cycle
Corps

The Hamiltonian characteristic of a graph requires that all the vertices are involved only once in the chain or cycle.

Content
Corps

To complete a car rally, competitors must pass through each of the milestones identified by letters on the following map.

Image
Graph of a rally route.
Corps

Bearing in mind that they must return to the starting point identified by the vertex |A| to finish the race, what could be one of the routes taken by the competitors?

Solution
Corps

We need to ensure that the start and end vertex are the same, while passing through each vertex only once.

So one possible route could be the following Hamiltonian cycle. ||A-E-D-C-F-B-A||

Text

See also

Links
Title (level 3)
Types of Graphs
Title slug (identifier)
types-of-graphs
Corps

Depending on the information provided on the graph, it can be given a specific name.

  • Related : when all the vertices are accessible from any vertex.

  • Tree : it's a graph that has no simple cycles.

  • Oriented : when the edges suggest an accurate orientation by means of an arrow.

  • Weighted (Valued) : when each of the edges has a quantity associated with it.

  • Coloured  : when each of the edges has a quantity associated with it.

Content
Columns number
3 columns
Format
33% / 33% / 33%
First column
Legend
English image coming soon!
Image
A connected graph and an unconnected graph.
Legend
English image coming soon!
Image
A weighted graph.
Second column
Legend
English image coming soon!
Image
A connected graph and an unconnected graph.
Legend
English image coming soon!
Image
An oriented graph.
Third column
Legend
English image coming soon!
Image
A tree graph.
Legend
English image coming soon!
Image
A coloured graph.
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See also

Links
Title (level 3)
The Critical Path
Title slug (identifier)
the-critical-path
Corps

In practical terms, the critical path is often used to establish a deadline for completing a project.

Content
Corps

Before you buy a home, it's important to analyse a number of factors that will help you make a wise purchase. To make sure you don't forget anything in the process, here are a few tips.

Tasks

Time (Days)

Prerequisites

A : Establishing your needs

1

None

B : Drawing up a budget

5

A

C : Shopping for a mortgage

7

B

D : Hiring a notary

3

B

E : Hiring a real estate agent

3

B

F : Visiting homes

182

C - D - E

G : Negotiating an interest rate

7

F

H : Obtain a pre-authorised loan

30

F

I : Making an offer

7

G - H

J : Have the home inspected

14

I

K : Signing the deed of sale

2

J

L : Taking out home insurance

10

J

M : Moving

1

K - L

What is the total duration of such a project?

Solution
Corps
  1. Construct a weighted (valuated) graph of the situation

Image
A weighted graph.
Corps
  1. Identify the path with the highest weight

Image
A weighted graph containing one critical path.
Corps
  1. Calculate the weight for this path

||\begin{align}
\text{Weight}&=1+5+7+182+30+7+14+10+1\\
\text{Weight}&=257
\end{align}||

Answer : It will take a total of |257| days to complete this project.

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See also

Links
Title (level 3)
The Chromatic Number
Title slug (identifier)
the-chromatic-number
Corps

In practical terms, the chromatic number is often used to colour a world map, design electronic chips or plan a telecommunications network.

Content
Corps

Keen to get to know her colleagues as well as possible, Ms Dreau wants to take part in as many of the activities offered by her school as possible. However, certain constraints in her timetable prevent her from taking part in everything she would like to.

  • journalism clashes with a few improvisation and success support sessions;

  • it's impossible for her to enrol in basketball, theatre and dance at the same time;

  • success support and theatre are both on the Monday evening timetable.

What is the maximum number of activities she can take part in?

Solution
Corps
  1. Draw a graph whose edges link incompatible elements.

Image
An activity graph.
Corps
  1. Assign different colours to adjacent vertices using a minimum number of colours.

Image
A coloured activity graph
Corps

Answer: Taking all these constraints into account, Ms Dreau will be able to take part in 3 activities: dance, success support and improvisation.

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See also

Links
Title (level 3)
The value tree
Title slug (identifier)
the-value-tree
Corps

In practical terms, the value tree is often used to minimise or maximise costs or distances.

Content
Corps

Before building homes in a new neighbourhood, a city must install a water and sewer system that connects each residence. Despite a few geographical constraints, most homes can be connected by this future system.

Image
A weighted (valuated) graph.
Corps

Taking the quantities in the graph as the distance, in metres, between each of the houses, what would be the minimum length of the network in this neighbourhood?

Solution
Corps
  1. Put the edges in ascending order according to their respective values

    |DF=199,| |DE=213,| |CE=256,| |AB=298,| |BD=332,| |BC=375,| |EF=395,| |AF=405,| |AC=657|

  2. Trace the graph again, placing the edges one by one, following the order established above, until all the vertices in the graph are connected.

Image
A weighted tree graph.
Corps

Answer: The minimum length for this network will be |298+332+199+213+256=1\ 298| metres.

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Links
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