18.4: Fair Division - Mathematics

1. Chance values the veggie half at $7.50 and pepperoni half at $2.50.

A full pepperoni slice is (frac{1}{4}) of the pepperoni half. Value ($ 2.50 / 4=$ 0.625)

A full veggie slice is (frac{1}{4}) of the veggie half. Value ($ 7.50 / 4=$ 1.875)

A slice that is ½ pepperoni (frac{1}{2}) veggie is value ($ 0.3125+$ 0.9375=$ 1.25)

3. Erin: Bowl 1, Catherine: Bowl 2, Shannon: Bowl 3

5. a. 25 Snickers @ $0.01 each, 20 Milky Ways @ $0.05 each, 60 Reese’s @ $0.02 each

Value: ($ 0.25+$ 1.00+$ 1.20=$ 2.45)

b. No. Dustin values the whole bag at $8, so a fair share would be $4.

c. Lots of possibilities. Here’s a couple:

80 Milky Ways, 0 Snickers, 0 Reese’s

50 Snickers, 50 Milky Ways, 50 Reese’s

7. Zoe

b. Maggie: s2, s3. Meredith: s1, s2. Holly: s3

c. Maggie: s2, Meredith: s1, Holly: s3, Zoe: s4

9. P5

b. $6.50 (doesn’t need to trim it much since they’re last)

c. P4 would receive it, with value $6.00 (since P4 would trim it)

11. ((320+220) / 4=$ 135)

b. Desk and Vanity both go to A. A pays ($ 320+$ 220-$ 135=$ 405) to estate

B gets $95, C gets $125, D gets $110.

c. Surplus of ($ 405-$ 95-$ 125-$ 110=$ 75) gets split, $18.75 each.

A gets desk and vanity, pays $386.25 to estate

B gets $113.75, C gets $143.75, D gets $128.75

13. Fair shares: Abby: 10.333, Ben: 9, Carla: 7.667

Motorcycle to Abby, Car to Ben, Tractor to Abby, Boat to Abby

Initial: Abby pays $10.667, Ben pays $2, Carla gets $7.667

Surplus: $5; $1.667 each

Final: Abby gets Motorcycle, Tractor and Boat, pays $9

Ben gets Car, pays $0.333

Carla gets $9.334

15. Fair shares: Sasha: $135, Megan: $140

Sasha gets: Couch, detail cleaning. Value $80

Megan gets: TV, Stereo, carpets. Value: $260

Initial: Sasha gets $55, Megan pays $120.

Surplus: $65; $32.50 each

Final: Sasha gets Couch and does detail cleaning, gets $87.50

Megan gets TV and stereo, and cleans carpets, pays $87.50

17. s3, worth $270

b. s1 and s4 have combined value $440 for Greedy, so piece would be worth $220

Go Math Grade 5 Answer Key Chapter 8 Divide Fractions

Students can grab the complete knowledge on Divide Fractions on Go Math Grade 5 Answer Key Chapter 8. This article consists of the solutions to practice problems, mid-chapter, and review tests along with answers and explanations for students to have more practice. So, the students who are in search of Go Math Grade 5 Answer Key can Download pdf from here.

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Fair Shares

This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework.
A PDF of the student activity is included.

Click on the image to enlarge it. Click again to close. Download PDF (1237 KB)

solve division problems using proportional adjustment

Number Framework Links
This activity can be used to encourage students at stage 7 to extend their range of strategies to include simplification by proportional adjustments.

FIO, Levels 3-4, Multiplicative Thinking, Fair Shares, pages 6-7

This activity deals with partitive division, that is, division where the number of parts is known but the amount of each share is not. Students learn that they can turn a complex division problem into a simpler one by simplifying both the dividend (the starting number) and the divisor (the number doing the dividing), using a common factor.
You could introduce the mathematics of question 1 with a simple example like the one below to illustrate the number properties. By doing this, you move the focus from the answer, which the students already know or can work out easily, to the number properties involved. By keeping the numbers manageable, you can use materials to demonstrate the transformations on the quantities involved.
Suppose that there are 32 biscuits to be shared among 8 people. How can the numbers in the problem be altered without affecting the size of the share?

Halving both the number of biscuits and the number of people who share them keeps the size of the shares the same: 32 ÷ 8 = 4, 16 ÷ 4 = 4, and 8 ÷ 2 = 4. The same is true no matter what number the numbers of biscuits and people are divided by. The size of the shares is also unaffected if the numbers of biscuits and people are doubled, trebled, or multiplied by any number at all, although this operation will seldom make a division problem easier. Algebraically, the principle can be expressed in this way: if a ÷ b = c, then a/n ÷ b/n = c where n is any number except zero.
Students will be able to use this method as long as they can spot a value for n that works, that is, a value that is a common factor of both numbers. For example, in 198 ÷ 6 =, students won’t be able to reduce the problem to 99 ÷ 3 = unless they realise that both 198 and 6 are even numbers and so have a common factor of 2.
Sometimes, it will take too much work to simplify a division problem using this method and another strategy is more suitable. Students need to recognise when halving, thirding, and so on of both numbers is an effective strategy and when it is not. Effectiveness depends on how easy it is to find a common factor and how easy it is to divide by that common factor. Take, for example, 657 ÷ 9 =. Both 657 and 9 are divisible by 3, but the difficulty of dividing 657 by 3 means that, for many students, standard place value methods are likely to be a better choice. For example: 630 ÷ 9 = 70 27 ÷ 9 = 3 70 + 3 = 73.
In question 1, the unknown is the total number of biscuits on the table in question 2, it is the size of each share. It is important that students meet division problems posed in different ways like this so that they come to understand the principle of inverse or reversibility.
Students need to be able to process question 3 without first solving the problem posed at the beginning: 168 ÷ 24 =. That is, they need to work backwards through a number of equivalent statements:

In this way, they operate on statements of equality while accepting the lack of closure that comes from not knowing the quotient (answer). They know that must be half of 168 because 12 is half of 24, and so on. This is the same kind of thinking that underpins equivalent fractions, for example,
The problems in question 4 can be solved in similar ways, but because the numbers are deliberately larger, the students will have to find the factors of the divisors (36, 28, and 27) before they can work backwards.
For question 5, the students should be very sure that their equations can be sensibly solved by strategies such as halving and doubling, place value, or working backwards, before they try them out on a classmate.

Answers to Activity

1. The completed equations are:
20 ÷ 4 = 5,
so 40 ÷ 8 = 5,
so 60 ÷ 12 = 5,
so 30 ÷ 6 = 5,
so 15 ÷ 3 = 5.
The pattern in these division equations is that doubling or halving both numbers produces the same answer. (60 ÷ 12 = 5, from 20 ÷ 4 = 5, is trebling.)
2. a. Grace used halving of both numbers to go directly from 72 ÷ 18 = 4 to 36 ÷ 9 = 4. Some other divisions using doubling and trebling patterns and basic multiplication
facts are: 12 ÷ 3 = 4, 24 ÷ 6 = 4, 48 ÷ 12 = 4, and 144 ÷ 36 = 4.
b. They will all have the same answer because both numbers in each division statement are either doubles or halves of other division statements that have an answer of 4. Sharing half as many biscuits among half as many people gives the same share.
3. 168 ÷ 24 = 84 ÷ 12 = 42 ÷ 6 = 7 biscuits per person
so 12 x 7 = 84 (at the 12-seat table)
and 6 x 7 = 42 (at the 6-seat table)
4 x 7 = 28 (at the 4-seat table),
and 3 x 7 = 21 (at the 3-seat table) or 42 ÷ 6 = 7, so 84 ÷ 12 = 7, 21 ÷ 3 = 7,
and 28 ÷ 4 = 7.
4. a. 11 biscuits. (396 ÷ 36 = 132 ÷ 12 = 66 ÷ 6 = 11)
b. 8 biscuits. (224 ÷ 28 = 112 ÷ 14 = 56 ÷ 7 = 8)
c. 9 biscuits. (243 ÷ 27 = 81 ÷ 9 = 9)
5. Problems will vary

Numeracy Project materials (see Numeracy Books page)
• Book 6: Teaching Multiplication and Division
Royal Cooking Lesson

Figure It Out
• Number Sense and Algebraic Thinking: Book Two, Level 3
Horsing Around, page 11


Previous qualifications Edit

Before the introduction of GCSEs, students took CSE (Certificate of Secondary Education) or the more academically challenging O-Level (General Certificate of Education (GCE) Ordinary Level) exams, or a combination of the two, in various subjects. The CSE broadly covered GCSE grades C-G or 4–1, and the O-Level covered grades A*-C or 9–4, but the two were independent qualifications, with different grading systems. The separate qualifications were criticised for disadvantaging the bottom 42% of O-Level entrants who failed to receive a qualification, and the highest-achieving CSE entrants who had no opportunity to demonstrate higher ability. [ citation needed ]

In its later years, O-Levels were graded on a scale from A to E, with a U (ungraded) grade below that. Before 1975, the grading scheme varied between examination boards, but typically there were "pass" grades of 1 to 6 and "fail" grades of 7 to 9. However the grades were not displayed on certificates. [ citation needed ]

The CSE was graded on a numerical scale from 1 to 5, with 1 being the highest, and 5 being the lowest passing grade. Below 5 there was a U (ungraded) grade. The highest grade, 1, was considered equivalent to an O-Level C grade or above, and achievement of this grade often indicated that the student could have taken an O-Level course in the subject to achieve a higher qualification. As the two were independent qualifications with separate syllabi, a separate course of study would have to be taken to "convert" a CSE to an O-Level in order to progress to A-Level. [ citation needed ]

There was a previous attempt to unite these two disparate qualifications in the 1980s, with a trial "16+" examination in some subjects, awarding both a CSE and an O-Level certificate, before the GCSE was introduced. The final O-level/CSE examinations were sat in 1987. [ citation needed ]

Introduction of the GCSE Edit

GCSEs were introduced in 1988 [2] to establish a national qualification for those who decided to leave school at 16, without pursuing further academic study towards qualifications such as A-Levels or university degrees. They replaced the former CSE and O-Level qualifications, uniting the two qualifications to allow access to the full range of grades for more students. However the exam papers sometimes had a choice of questions designed for the more able and the less able candidates.

Upon introduction, the GCSEs were graded on a letter scale, from A to G, with a C being set as roughly equivalent to an O-Level Grade C, or a CSE Grade 1, and thus achievable by roughly the top 25% of each cohort.

Changes since initial introduction Edit

Over time, the range of subjects offered, the format of the examinations, the regulations, the content, and the grading of GCSE examinations has altered considerably. Numerous subjects have been added and changed, and various new subjects are offered in the modern languages, ancient languages, vocational fields, and expressive arts, as well as Citizenship courses. [3]

Introduction of the A* grade Edit

In 1994, the A* grade was added above the grade A, to further differentiate attainment at the very highest end of the qualification. This remained the highest grade available until 2017. The youngest pupil to gain an A* grade was Thomas Barnes, who earned an A* in GCSE Mathematics at the age of 7. [4]

2000s reforms Edit

Between 2005 and 2010, a variety of reforms were made to GCSE qualifications, including increasing modularity and a change to the administration of non-examination assessment.

From the first assessment series in 2010, controlled assessment replaced coursework in various subjects, requiring more rigorous exam-like conditions for much of the non-examination assessed work, and reducing the opportunity for outside help in coursework.

2010s reforms Edit

Under the Conservative government of David Cameron, and Education Secretary Michael Gove, various changes were made to GCSE qualifications taken in England. Before a wide range of reforms, interim changes were made to existing qualifications, removing the January series of examinations as an option in most subjects, and requiring that 100% of the assessment in subjects from the 2014 examination series is taken at the end of the course. These were a precursor to the later reforms. [5]

From 2015, a large-scale programme of reform began in England, changing the marking criteria and syllabi for most subjects, as well as the format of qualifications, and the grading system. [6] [7]

Under the new scheme, all GCSE subjects were revised between 2015 and 2018, and all new awards will be on the new scheme by summer 2020. The new qualifications are designed such that most exams will be taken at the end of a full 2-year course, with no interim modular assessment, coursework, or controlled assessment, except where necessary (such as in the arts). Some subjects will retain coursework on a non-assessed basis, with the completion of certain experiments in science subjects being assumed in examinations, and teacher reporting of spoken language participation for English GCSEs as a separate report.

Other changes include the move to a numerical grading system, to differentiate the new qualifications from the old-style letter-graded GCSEs, publication of core content requirements for all subjects, and an increase in longer, essay-style questions to challenge students more. Alongside this, a variety of low-uptake qualifications and qualifications with significant overlap will cease, with their content being removed from the GCSE options, or incorporated into similar qualifications. A range of new GCSE subjects were also introduced for students to study from 2017, 2018. 2019, and 2020. [8]

GCSE examinations in English and mathematics were reformed with the 2015 syllabus publications, with these first examinations taking places in 2017. The remainder were reformed with the 2016 and 2017 syllabus publications, leading to first awards in 2018 and 2019, respectively.

For GCSE Science, the old single-award "science" and "additional science" options are no longer available, being replaced with a double award "combined science" option (graded on the scale 9–9 to 1–1 and equivalent to 2 GCSEs). Alternatively, students can take separate qualifications in chemistry, biology, and physics. Other removed qualifications include a variety of design technology subjects, which are reformed into a single "design and technology" subject with multiple options, and various catering and nutrition qualifications, which are folded into "food technology". Finally, several "umbrella" GCSEs such as "humanities", "performing arts", and "expressive arts" are dissolved, with those wishing to study those subjects needing to take separate qualifications in the incorporated subjects. [9]

Implications for Wales and Northern Ireland Edit

These reforms do not directly apply in Wales and Northern Ireland, where GCSEs will continue to be available on the A*-G grading system. However, due to legislative requirements for comparability between GCSEs in the three countries, and allowances for certain subjects and qualifications to be available in Wales and Northern Ireland, some 9–1 qualifications will be available, and the other changes are mostly adopted in these countries as well. [10]

In Northern Ireland, a decision was taken by Minister of Education, Peter Wier (DUP), in 2016 [11] to align the A* Grade to the 9 Grade of the English reformed qualifications. The first award of the new A* grade being in 2019. A C* grade was also introduced in Northern Ireland to align to the 5 Grade in England, again with first awarding in 2019. GCSEs in Northern Ireland remain modular and science practicals can count towards the overall grade outcome. Speaking and listening also remains a component of the GCSE English Language specification.

Historically, there were a variety of regional examination boards, or awarding organisations (AOs), who set examinations in their area. Over time, as deregulation allowed schools to choose which boards to use, mergers and closures have led to only 5 examination boards remaining today:

    (AQA), which absorbed the following boards: AEB, JMB, NEAB, and SEG. (OCR), which absorbed the Oxford Delegacy of Local Examinations, Cambridge Local Examinations, Oxford & Cambridge Examinations Board, MEG, and RSA exam boards. , which absorbed the LREB, BTEC, and ULEAC boards. (WJEC or CBAC), the main examining board in Wales. (CCEA), the examining board and regulator in Northern Ireland.

The examination boards operate under the supervision of Ofqual (The Office of Qualifications and Examinations Regulation) in England, Qualifications Wales in Wales, and the CCEA in Northern Ireland.

In England, AQA, OCR, and Pearson operate under their respective brands. Additionally, WJEC operate the brand Eduqas, which develops qualifications in England. CCEA qualifications are not available in England.

In Wales, WJEC is the only accredited awarding body for GCSEs in the public sector, and thus no other board formally operates in Wales. However, some qualifications from the English boards are available as designated qualifications in some circumstances, due to not being available from WJEC.

In Northern Ireland, CCEA operates as both a board and a regulator. Most qualifications from the English boards are also available, with the exception of English language and the sciences, due to requirements for speaking and practical assessment, respectively. [12]

Students usually take at least 5 GCSEs in Key Stage 4, in order to satisfy the long-standing headline measure of achieving 5 A*-C grades, including English, Mathematics, and Science. The exact qualifications taken by students vary from school to school and student to student, but schools are encouraged to offer at least one pathway that leads to qualification for the English Baccalaureate, requiring GCSEs in English language, English literature, mathematics, 2 science GCSEs, a modern or ancient language, and either history or geography. [ citation needed ]

Subjects Edit

The list of currently available GCSE subjects is much shorter than before the reforms, as the new qualifications in England all have core requirements set by the regulator, Ofqual, for each subject. In addition, there are several subjects where only one board offers qualifications, including some that are only available in one country of the UK for that reason. The following lists are sourced from the exam board websites. [13] [14] [15] [16] [17] [18] [19] [20]

Core subjects Edit

These are the requirements for achieving the English Baccalaureate headline measure in league tables, from 2017 onwards. [21] The Baccalaureate itself does not garner a certificate for students. Other subjects, especially religious studies, citizenship studies, computer science, or physical education are compulsory in some schools as these subjects form part of the National Curriculum at Key Stage 4.

  • English: both English Language and English Literature
  • Mathematics
  • Science: See GCSE Science. Biology, Chemistry and Physics or Combined Science. Computer Science is also considered a science for the English Baccalaureate
  • Languages: one GCSE in a modern or ancient language
    • Modern languages: Arabic, Bengali, Chinese (Cantonese), Chinese (Mandarin), French, German, Modern Greek, Gujarati, Modern Hebrew, Italian, Japanese, Panjabi, Persian, Polish, Portuguese, Russian, Spanish, Turkish, Urdu
    • Ancient languages: Classical Greek, Biblical Hebrew, Latin

    Other subjects Edit

    • Sciences and Mathematics:
      • Astronomy
      • Geology
      • Psychology
      • Statistics
      • Sociology
      • Ancient History
      • Citizenship Studies
      • Classical Civilisation
      • Religious Studies
      • Business Studies
      • Economics
      • Design and Technology
      • Electronics
      • Engineering
      • Food Preparation & Nutrition
      • Art and Design
      • Dance
      • Drama
      • Film Studies
      • Media Studies
      • Music
      • Photography
      • Graphics
      • Physical Education
      • Agriculture and Land Use
      • Business and Communication Systems
      • Child Development
      • Construction and the Built Environment
      • Contemporary Crafts
      • Digital Technology
      • Further Mathematics
      • Government and Politics
      • Health and Social Care
      • Home Economics
      • Hospitality
      • Irish
        • Irish
        • Gaeilge
        • Information and Communication Technology
        • Welsh (compulsory in Welsh schools):
          • Welsh Language (first language)
          • Welsh Literature (first language)
          • Welsh Second Language

          Grades and tiering Edit

          GCSEs are awarded on a graded scale, and cross two levels of the Regulated Qualifications Framework (RQF): Level 1 and Level 2. These two levels roughly correspond, respectively, to foundation and higher tier in tiered GCSE qualifications. Level 1 qualifications constitute GCSEs at grades G, F, E, and D or 1, 2, and 3. Level 2 qualifications are those at grades C, B, A, and A* or 4, 5, 6, 7, 8, and 9.

          The tiering of qualifications allows a subset of grades to be reached in a specific tier's paper. Formerly, many subjects were tiered, but with the mid-2010s reform, the number of tiered subjects reduced dramatically, including the removal of tiering from the GCSE English specifications. Untiered papers allow any grade to be achieved. Coursework and controlled assessment tasks are always untiered.

          In the past, mathematics qualifications offered a different set of tiers, with three. These were foundation tier at grades G, F, E, and D intermediate tier at grades E, D, C, and B and higher tier at grades C, B, A, and A*. This eventually changed to match the tiers in all other GCSE qualifications.

          The evolution of grades, and a rough comparison between them is as follows:

          Approximate equivalences for GCSE, O-Level and CSE grades
          GCSE Grade O-Level Grade CSE Grade
          from 2017 a
          Northern Ireland
          from 2019 b
          Wales from 1994
          England, NI 1994–2019 c
          1988–1993 1975–1987 d 1965–1987
          9 A* A* A A
          8 A
          6 B B B B
          5 C*
          C C C 1
          4 C
          3 D D D D 2
          E E E E 3
          F F F U 4
          G G G 5
          U U U U U

          • Notes:
            • GCSE grades 9 to 4 (A* to C) – Certificate and qualification awarded. At GCSE, considered a 'good pass', and awards a qualification at Level 2 of the RQF.
            • GCSE grades 3 to 1 (D to G) – Certificate and qualification awarded. At GCSE, awards a qualification at Level 1 of the RQF.
            • U: ungraded/unclassified – no certificate or qualification awarded
            • ^a 9–1 grades phased in by subject between 2017 and 2019 in England
            • ^b New A*–G grades in Northern Ireland from 2019 [22]
            • ^c A*–G grades as used in Wales since 1994, and in England and Northern Ireland between 1994 and 2019
            • ^d Before 1975, each exam board had its own grading system (some used letters, others numbers). Grades were only given to schools and not recorded on students' certificates

            Letter grades Edit

            When GCSEs were first introduced in 1988, they were graded on a letter scale in each subject: A, B, C, D, E, F, and G being pass grades, with a U (unclassified) grade below that which did not qualify the student for a certificate.

            These grades were initially set such that a GCSE grade C was equivalent to an O-Level grade C or a CSE grade 1, though changes in marking criteria and boundaries over the years mean that this comparison is only approximate.

            Infrequently, X and Q grades are awarded. X indicates that a course was not completed in full, and therefore an appropriate grade cannot be calculated. The Q (query) grade is a temporary grade that requires the school to contact the examining body. These latter two grades are both usually provisional, and are replaced with a regular grade once any issues have been resolved. X grades are also sometimes used for other purposes, on rare occasions, such as to indicate that an examiner found offensive material or hate speech within a student's responses. In some cases, this may lead to the student losing all marks for that paper or course. These grades are most common in subjects which discuss ethical issues, such as biology, religious studies, and citizenship. [ citation needed ]

            In 1994, an A* grade was added above the initial A grade to indicate exceptional achievement, above the level required for the A grade.

            Under the letter grade scheme, foundation tier papers assess content at grades C to G, while higher tier papers assess content at grades A* to C. In foundation tier papers, the student can obtain a maximum grade of a C, while in a higher tier paper, they can achieve a minimum grade of a D. If a higher tier candidate misses the D grade by a small margin, they are awarded an E. Otherwise, the grade below E in these papers is U. In untiered papers, students can achieve any grade in the scheme. [ citation needed ]

            This scheme is being phased out in England, but remains in Wales and Northern Ireland. In Northern Ireland, the A* grade has been adjusted upwards with the introduction of the numerical scheme in England, such that an A* is equivalent to a new English grade 9. Northern Ireland also added a C* grade to line up with the grade 5 in the English grading. [ citation needed ]

            Numerical grades (2017 onwards) Edit

            From 2017 in England (and in Wales and Northern Ireland on qualifications from the English-based awarding bodies), most GCSEs are now assessed on a 9-point scale, using numbers from 9 to 1, and, like before, a U (unclassified) grade for achievement below the minimum pass mark. Under this system, 9 is the highest grade, and is set above the former A* classification, equivalent to the new Northern Irish A* grade. The former C grade is set at the new grade 4, now known as a "standard pass", and grade 5 being considered a "strong pass" under the new scheme.

            Although fewer qualifications have tiered examinations than before, the tiering system still exists. At foundation tier, the grades 1, 2, 3, 4, and 5 are available, while at higher tier, the grades 4, 5, 6, 7, 8, and 9 are targeted. Once again, if a higher-tier student misses the grade 4 mark by a small margin, they are awarded a grade 3. Controlled assessment and coursework tasks are untiered.

            The youngest person known to have achieved a grade 9 is Ellie Barnes who achieved the grade in Mathematics aged 8 years old. [23] [24] [25]

            Results Edit

            GCSE results are published by the examination board in August, for the previous exam series in April to June of the same year. They are usually released one week after the A-Level results, on the Thursday which falls between 20 August and 26 August. The examination results are released to centres (schools) prior to the release to candidates and the public. Examination results are released by the Joint Council for Qualifications (JCQ), which represents the main GCSE awarding organisations. Some boards and schools release results online, although many still require students to attend in person to collect their results from the centre they sat exams at. [26]

            In England, these results then go on to inform league tables published in the following academic year, with headline performance metrics for each school.

            UK GCSE Grades Awarded (%'age) (letter system) [27]
            A* A B C D E F G U A*+A A*-C entries
            1988 N/A 8.4 12.8 20.7 19.3 16.6 12.5 6.3 3.4 8.4 41.9 5,230,047
            1989 9.9 13.8 21.9 19 15.8 11.2 5.6 2.9 9.9 45.6 5,132,998
            1990 10.8 14.4 22.5 18.7 15.3 10.6 5.2 2.5 10.8 47.7 5,016,547
            1991 11.4 14.7 22.4 18.6 15 10.5 5.3 2.2 11.4 48.5 4,947,593
            1992 12.3 15.3 22.9 18.6 14.7 9.9 4.7 1.6 12.3 50.5 5,028,554
            1993 12.5 15.9 23.1 18.6 14.2 9.3 4.4 1.8 12.5 51.5 4,968,634
            1994 2.8 10.2 18 21.8 18.7 13.7 9.3 4.1 1.5 13 52.8 5,029,599
            1995 3.2 9.9 17.8 22.1 18.6 14 9 3.9 1.5 13.1 53 5,431,625
            1996 3.4 10.3 18 22.3 18.6 13.4 8.7 3.8 1.5 13.7 54 5,475,872
            1997 3.6 10.5 18.1 22.3 18.7 13.3 8.5 3.6 1.5 14.1 54.4 5,415,176
            1998 4.1 10.6 16.5 23.6 18.6 13.2 7.6 3.5 2.3 14.7 54.8 5,353,095
            1999 4.4 10.8 16.9 23.7 18.7 12.7 7.5 3.3 2 15.2 55.8 5,374,751
            2000 4.6 11.2 17 23.8 18.4 12.5 7.2 3.2 2.1 15.8 56.6 5,481,920
            2001 4.9 11.2 16.9 24.1 18.3 12.1 7.1 3.3 2.1 16.1 57.1 5,632,936
            2002 5 11.4 17.4 24.1 18.1 12 6.7 3.2 2.1 16.4 57.9 5,662,382
            2003 5.1 11.6 17.3 24.1 17.7 11.7 6.8 3.3 2.4 16.7 58.1 5,733,487
            2004 5.6 11.8 17.3 24.5 17.3 11.3 6.6 3.2 2.4 17.4 59.2 5.875,373
            2005 5.9 12.5 18 24.8 17.3 10.5 6 2.8 2.2 18.4 61.2 5,736,505
            2006 6.3 12.8 18.3 25 17.3 10.2 5.6 2.6 1.9 19.1 62.4 5,752,152
            2007 6.4 13.1 18.6 25.2 17.2 9.8 5.3 2.4 2 19.5 63.3 5,827,319
            2008 6.8 13.9 19.8 25.2 16.6 9.1 4.7 2.3 1.6 20.7 65.7 5,669,077
            2009 7.1 14.5 19.9 25.6 16.5 8.5 4.4 2.1 1.4 21.6 67.1 5,469,260
            2010 7.5 15.1 20.6 25.9 15.9 7.8 4 1.9 1.3 22.6 69.1 5,374,490
            2011 7.8 15.4 21.7 24.9 15.1 7.8 4.1 2 1.2 23.2 69.8 5,151,970
            2012 7.3 15.1 21.7 25.3 15.9 7.7 4.1 1.9 1 22.4 69.4 5,225,288
            2013 6.8 14.5 21.5 25.3 16.6 8 4.1 2 1.2 21.3 68.1 5,445,324
            2014 6.7 14.6 21.9 25.6 16.3 7.6 3.8 2.0 1.5 21.3 68.8 5,217,573
            2015 6.6 14.6 22.1 25.7 16.4 7.6 3.7 1.9 1.4 21.2 69 5,277,604
            2016 6.5 14.0 21.4 25.0 16.9 8.3 4.2 2.1 1.6 20.5 66.9 5,240,796
            2017 7.1 14.2 20.6 23.5 16.8 9.3 4.7 2.3 1.5 21.3 65.3 3,694,771
            2018 7.0 14.7 21.8 23.4 15.2 8.5 4.5 2.7 2.2 21.7 66.9 860,246

            Source: Joint Council for General Qualifications via Brian Stubbs.
            Note: In the final year DES statistics for O-Levels are available, and across all subjects, 6.8% of candidates obtained a grade A, and 39.8% achieved grades A to C.

            70 Hilarious Math Jokes For Kids

            1. Not So Smart Sheepdog

            One day, a sheepdog decides to help his farmer by getting all the sheep into the pen. After sending the sheep into the pen, he returns back to the farm to inform the farmer that all 40 sheep have been sent safely to their haven.

            The farmer says, “There are just 36 instead of 40. Just now I counted them”.

            To this, the sheepdog replies, “Yes, I know. I just rounded them up for you”.

            2. One Of The Best Wordplays On Math

            Q. How do you find the best math tutor in the city?

            3. We Can Bet Even The Best Of Mathematicians Won’t Be Able To Answer This Question

            Q. How many sides do you find in a circle?

            4. Fair Enough For The Dog

            Q. Why did the dog cross the Mobius strip?

            A. Because he wanted to get to the same side.

            5. Now That’s An Obedient Student

            Q. Why did the math student do his homework on the floor?

            A. Because his teacher instructed him not to use tables.

            6. At Least One Monster Is Good At Mathematics

            Q. Which monster is good at math?

            A. No one actually, unless you Count Dracula!

            7. Only The Smartest Will Get It

            There are ten types of people in the world. Those who understand binary and those who don’t.

            8. Yes, Even Snakes Are Good At Math!

            Q. Which type of snake in best at math?

            9. Someone’s Mad!

            Q. What did one algebra book say to the other?

            A. Please do not bother me right now. I’ve to deal with my own problems.

            10. Definitely Trying It This Winter:

            Q. What is the best way to keep warm in a square room?

            A. You huddle right into the corner, where it’s always 90 degrees.

            11. Isn’t It A Favorite Of Humans As Well?

            Q. What is the most favorite type of math of birds?

            12. That’s The Kind Of Reassurance We Need

            Q. What did the calculator say to the student?

            A. You can always count on me.

            13. Another Excellent Math Pun:

            Q. What tool does a mathematician use to plow a field?

            14. Well, Nothing Wrong In Being A Fitness Freak!

            Q. Why did a circle do a flip?

            A. Because he wanted to get in shape.

            15. Math Riddle For Kids:

            Q. If two is a company and three is a crowd, what are four and five?

            16. Another Math Riddle To Crack You Up!

            Q. On a fine spring Sunday, two dads and two sons decide to go fishing. Luck strikes and each of them catch one fish. But they bring just three fish home? Can you tell why?

            A. Because a grandfather, his son and his son’s son went fishing. Hence, there were just three people.

            17. He’s Not Completely Wrong!

            If I had 7 apples in one hand and 8 oranges in another, what would I have?

            18. Halloween Math Joke For Kids:

            Q. If you divide the circumference of a Jack-O-Lantern by its diameter, what would you get?

            19. The Best Way To Improve Your Math!

            Why did the student wear spectacles during math class?

            Because it would improve di-vision.

            20. Not A Very Delicious Cake, unfortunately!

            Q. Why did the girl eat her math homework?

            A. Because her math teacher told her that it was just a piece of cake.

            21. We Can Feel The Plot Thickening

            Do not trust a math teacher holding a graph paper. They could be plotting something.

            22. There’s Always Someone In The Group

            Q. What do you call friends who love Mathematics?

            23. Yes, Even Numbers Tend To Wander!

            Q. What is the term for numbers that always wander?

            24. Even Numbers Have Lunch

            Q. Why did the two 4s refuse to have lunch?

            A. Because they already 8 (ate)!

            25. Christmas Math Joke For Kids

            Q. Why is an artificial Christmas tree like the fourth root of -68?

            A. Because neither of the two has real roots!

            26. The Voice Mail Of A Math Professor

            Q. What are you most likely to hear in the voicemail of a Math professor?

            A. “The number you have dialed does not exist. Please, rotate your phone by 90 degrees and then try again.

            27. Indeed It Is (Not)

            You should never really let advanced math intimidate you. Because It’s as easy as pi!


            J.A. Fair High School was established in 1981, [ citation needed ] with construction completed and doors open to students in August 1982. [3] The school was named for James Augustus Fair, an educator, who spent his career as a biology teacher, administrator and after retirement served on the Pulaski County School Board.

            From its opening in 1982 through June 1987, FHS served as a junior/senior high school (grades 7-12) for the Pulaski County Special School District. In August 1987, FHS opened as solely a senior high school for the Little Rock School District, [3] one of 14 schools annexed to enhance desegregation efforts. FHS became a magnet school in the fall of 2000. With construction completed in the spring of 2004, the school now has three new labs for the academy programs, along with two new classrooms, a new band room, and an expanded cafeteria. [ citation needed ]

            In 2016 Michael Anthony became the principal. [4]

            Fair was replaced by a new high school in southwest Little Rock that began construction in 2017, [5] and opened as Little Rock Southwest High School in 2020. [6]

            The school features three magnet programs: Environmental Science, Information Science & Systems Engineering, and Medical Science, along with a Freshman Academy, High Schools That Work (HSTW), SECME. A variety of academic programs (which include 15 AP courses and a Community Based Instruction Program for students with moderate to severe disabilities), sports, club, and activity offerings.

            Three College and Career Academies: The Academy of Environmental Science, Enterprise Mobile Network Management Academy, and The Academy of Sports Medicine, along with a Freshman Academy. Academy-specific courses in technology, environmental studies, and sports medicine will drive the curriculum. Common groups of cross-disciplinary teachers will work with common groups of students throughout the academic year. All academy courses will target hands-on, project-based learning. The Business Industry will regularly interact with the students and teachers. Students will be required to create advanced senior projects designed to reflect high levels of college and career preparedness. We have an in-depth ongoing partnership with the University of Arkansas at Little Rock Department of Information Science via the Information Technology and E-Commerce Program and USDA. Year-round, regularly scheduled professional development will be provided to all teachers targeting learning goals aligned with technology. Students will be provided opportunities to gain industry certification and possible college credit.

            J A. Fair has also partnered with the Arkansas AIMS to strengthen the teaching of the AP® mathematics, science, and English courses and to build enrollment and increase the number of students taking and earning qualifying scores on AP® exams in these subjects.. A variety of academic programs (which include 15 AP courses and a Community Based Instruction Program for students with moderate to severe disabilities), sports, club, and activity offerings.

            The school mascot and athletic emblem is the War Eagle with the school colors of silver, blue (navy), and white.

            Athletics Edit

            Between the years of 2012–14, the J.A. Fair War Eagles participated in the 6A Classification within the 7A/6A South Conference as administered by the Arkansas Activities Association. Due to the enrollment count of J.A. Fair in the fall of 2014, they were reclassified within the 5A Central Conference for the 2014-2016 school years. The War Eagles competes in football, volleyball, cross country (boys/girls), bowling (boys/girls), swimming (boys/girls), basketball (boys/girls), soccer (boys only), baseball, softball, and track and field (boys/girls). [7]

            • Football: The War Eagles Football team won state football 5A championship in the fall of 1998, going on to defeat Cabot High School 41-0 continuing an undefeated season with a 14–0 record. The school placed 11 players on the Arkansas Democrat-Gazette's All Metro team following the victory led by Dameon Ashford, Tye Forte(QB), Faquan Harris, Gustavo Pena(K) and future NFL player Cedric Cobbs(RB).
            • Basketball: The War Eagles Basketball team won state basketball 5A championship in 2000 going on to defeat Fort Smith Southside 49-35 completing an undefeated season of 31–0. The War Eagles won another state championship in 2003 led by senior Vince Hunter who won the MVP of the Class 4A state tournament, he averaged 15.0 points, 12.0 rebounds and 8.0 blocked shots a game as a senior.
            • Cross Country: The War Eagles boys cross country team won a state cross country championship in fall 2001.
            • Tennis: The War Eagles boys tennis team won a state tennis championship in spring 1998.
            • Track and field: The War Eagles boys track team won a state track championship in spring 1999 led by All-American Nominee Kyle Cleveland.

            Clubs and traditions Edit

            Fair students may participate or be selected for a variety of clubs and organizations including Art Club, Band, Beta Club, Choir, FBLA, FCCLA, National Honor Society, Fire Marshals, Quiz Bowl, Student Council, and Yearbook.


            Born in San Francisco, California, Fair was the son of Robert A. Fair. [1] He attended St. Paul's Grammar School in San Francisco and graduated in 1941 from the High School of Commerce, where he was active in the school's Reserve Officer Training Corps program. [1] [4]

            Fair attended the University of San Francisco before receiving an appointment to the United States Military Academy. He entered West Point on July 19, 1942. He was appointed by Rep (R-CA) Richard J. Welch, the California 5th District. Found to be deficient in Mathematics, he was dismissed on January 12, 1943. [5] Prior to his enlistment in the Army, he also attended the University of Chicago from 1943 to 1944. [1]

            Fair attended Infantry Officer Candidate School at Fort Benning, Georgia, and received his commission as an Infantry officer in 1944. [1]

            His initial assignment was as an instructor in the Weapons Department at The Infantry School, Fort Benning. Following attendance at the University of Michigan for Japanese language training, First Lieutenant Fair served as an intelligence officer and interpreter on General MacArthur's Staff in Tokyo from 1946 to 1948. [4]

            He returned to the United States in August 1948, and was assigned to Fort Lewis, Washington, where he served as a platoon leader and regimental staff officer with the 23d and 38th Infantry Regiments. [1]

            Korean War Edit

            In August 1950, he accompanied the 38th Infantry Regiment to Korea. Fair served on the 38th Infantry Regimental Staff during the Korean War, where he fought on the Naktong River and later was to meet the Chinese Forces in North Korea he later became the G-3 (Air), 2nd Infantry Division. [1] Fair was awarded the Silver Star, Bronze Star, and Purple Heart medals and an Army Commendation Ribbon for meritorious service. [6] On November 29, 1950, Captain Fair earned the Silver Star for actions with Headquarters Company, 38th Infantry Regiment, in the vicinity of Kunu-ri. [7]

            Interwar service Edit

            Upon returning to the United States, he served as Assistant G-3 (Operations), Sixth U.S. Army at The Presidio of San Francisco in California. In August 1952, he attended the Infantry Officers' Advanced Course at Fort Benning, Georgia. Upon graduation, Captain Fair was assigned as the Regimental Operations Officer (and Ceremonial Officer) for the 3d Infantry Regiment (The Old Guard) at Fort Myer, Virginia, until August 1954. [1]

            After attending the Command and General Staff College at Fort Leavenworth, Kansas, as a Captain, Fair was assigned to Headquarters, V Corps, U.S. Army Europe as the Plans Officer, G-3/5/7.
            He attended the Armed Forces Staff College in Norfolk, Virginia, from August 1958 to January 1959. From there he served as a staff officer in the Joint War Plans Division, Office of the Deputy Chief of Staff for Military Operations from January 1959 to August 1960.

            His next assignment took him to Offutt Air Force Base, Nebraska, where he worked on the Joint Strategic Target Planning Staff as a Plans Officer. From August 1962 to June 1963, he attended the Naval War College in Newport, Rhode Island. Following this he became Secretary of the General Staff, Headquarters, Eighth U.S. Army from July 1963 to August 1964. [1]

            In August 1964, Lieutenant Colonel Fair assumed command of the 1st Battalion, 6th Infantry (Mechanized), 1st Armored Division at Fort Hood, Texas.

            He then returned to Washington, D.C., where he served as a member of the Department of the Army Board of Inquiry on the Army Logistics System the Chief of the Coordination Division, Office of the Chief of Staff, and subsequently as Executive Officer to the Vice Chief of Staff, U.S. Army. [1]

            Vietnam War Edit

            Colonel Fair commanded 1st Brigade, 25th Infantry Division in Tay Ninh, in South Vietnam, in 1968–1969. He received the Vietnamese Cross of Gallantry for his participation in heavy fighting around Ven Cau. During combat in Vietnam, his brigade was awarded the U. S. Valorous Unit Award and the Vietnamese Cross of Gallantry (with Palm). [1] He was instrumental in establishing a "Holiday Inn" in the Tay Ninh province so that soldiers returning from combat missions had an opportunity to shower, do laundry, repair their gear, and get a good night's sleep, a hot meal, and a little rest and recuperation (R & R). Fair was quoted, "We want to make the man who struggles in the field day and night, week after week, feel like a king for a couple of days." He later became the Chief of Staff, 25th Infantry Division, in Cu Chi, Vietnam.

            Cold War Edit

            Fair was assigned to the Management Information Systems Directorate, Office of the Assistant Vice Chief of Staff from September 1969 to April 1970. He later became Director of the Management Information Systems, and was promoted to Brigadier General. [1] Fair commanded the 2nd Armored Division at Fort Hood from July 16, 1973, to August 5, 1975. He worked diligently for operations-intelligence integration as the 2nd Armored Division prepared for its return of forces to Germany (Exercise Reforger) mission, and their annual Reforger exercise supporting the Army’s operational plans. During the Cold War, the 2nd Armored Division's primary mission was to prepare to conduct heavy armored combat against Warsaw Pact forces in defense of NATO. Hell On Wheels formed a key component of the U.S. military's plan to move "ten divisions in ten days" to Europe in the event of a Soviet threat to NATO. The division practiced this task numerous times during Exercise REFORGER (Return of Forces to Germany) from 1967 to 1988. [8]

            Josiah Bunting noted in his 1975 article on the Volunteer Army: "In fiscal 1974, Fair's division reenlistment goal was 813: 1222 took their burst of six. In July 1973, before Fair got to the division, the A.W.O.L. rate was 44 per 1000 a year later, it was 14 per 1000. From January to June 1974, 1194 troopers raised their G.T. scores, and of all the soldiers who re-upped when Fair was in command, 72 percent re-enlisted for his division." Bunting commented that Fair would be "a tough act to follow." [9]

            On August 25, 1975, Lieutenant General Fair assumed command of V Corps at Frankfurt, Federal Republic of Germany from Lieutenant General William R. Desobry. He was responsible for the training and readiness of V Corps, headquartered at Abrams Barracks, which included the 8th Infantry Division (Mechanized) "Pathfinder," the 3rd Armored Division "Spearhead," and the 11th Armored Cavalry Regiment "Blackhorse." Major General John R.D. Cleland, Jr. commanded the "Pathfinder" Division, which was headquartered at Rose Barracks, Bad Kreuznach. Major General (later Lieutenant General) Charles J. Simmons commanded the "Spearhead" Division headquartered at Frankfurt. Colonel (later Lieutenant General) John L. Ballantyne III commanded the "Blackhorse" Regiment headquartered at Fulda.

            While he was reportedly the only modern day corps commander to be relieved of duty during peacetime, [10] [11] a Time magazine article noted, "Lieut. General Robert L. Fair is headed for a more prosaic destination, however, and defenders of a tough, no nonsense, old-style Army are dismayed. As of next week Fair, 52, will retire for 'personal reasons' – the most important being that he and his commanding officer hated each other's guts." [12] General George S. Blanchard served as the USAREUR Commander from 1975-1979. In fact there is no evidence that either general did not respect the other. Truth of the matter, Gen Fair was most concerned about troop readiness and Gen Blanchard was concerned about his obligations to Congress. Both Generals were dedicated to service to the US Army. Gen Blanchard sent Gen Fair a back channel message advising him to return monies he had transferred from construction to training back to construction. Gen Fair did not do so. These funds were regulated by US Congress. [13] Lieutenant General (later General) Donn A. Starry assumed command of V Corps on February 16, 1976. [14]

            Following retirement from the military, Fair worked for SRI International, formerly Stanford Research Institute, before transitioning to Lockheed Corporation. While at Lockheed, he worked on the MQM-105 Aquila remote piloted vehicle (RPV), [15] now called unmanned aerial vehicles (UAV).

            Fair died on September 14, 1983, at Santa Clara, California. He is buried on The Presidio of San Francisco at San Francisco National Cemetery [16]

            Fair was married to the former Alys Tendrich on July 9, 1949. They had three children. [1]

            Major Lieutenant Colonel Colonel
            O-4 O-5 O-6
            July 10, 1956 June 13, 1961 October 12, 1966

            Brigadier General Major General Lieutenant General
            O-7 O-8 O-9
            June 1, 1970 May 1, 1972 August 5, 1975

            General Fair’s awards include the Silver Star, Legion of Merit with two Oak Leaf Clusters, Distinguished Flying Cross, Bronze Star with two Oak Leaf Clusters, Purple Heart with two Oak Leaf Clusters, Air Medal, Combat Infantryman’s Badge with Star (two awards), and Army Staff Identification Badge. His foreign awards include the Vietnam Distinguished Service Order (2nd Class), Vietnam Cross of Gallantry with Gold and Bronze Stars, the Vietnam Armed Forces Honor Medal (1st Class), and the Vietnam Psychological Operations Award (1st and 2nd Class). [1]


            Finally, let’s introduce the concept of independence, and two key theorems that deal with it.

            (A) is independent of (B) if (p(A given B) = p(A)) and (p(A) > 0) .

            Now we can state and prove the Multiplication rule.

            If (A) is independent of (B) , then (p(A wedge B) = p(A)p(B)) .

            Finally, we prove another useful fact about independence, namely that it goes both ways.

            Independence is Symmetric

            If (A) is independent of (B) , then (B) is independent of (A) .

            We’ve now established that the laws of probability used in this book can be derived from the three axioms we began with.

            3.1. Demographic Characteristics of the Participants

            3.2. Mental Health Knowledge Level among College Students

            3.3. Depressive Symptoms among College Students

            3.4. Associated Factors of Depressive Symptoms among College Students

            3.5. Association between Mental Health Knowledge Level and Depressive Symptoms among College Students

            0.98) and the 4th quartile (18

            0.99) were associated with a lowered risk of depressive symptoms, compared with MHKQ scoring in the 1st quartile (0

            15). After adjusting for personal associated factors such as family origins, physical status, sleep quality, and academic performance, college students with MHKQ scoring in the 3rd quartile (17

            0.98) and the 4th quartile (18

            0.91) were associated with a lowered risk of depressive symptoms, compared with MHKQ scoring in the 1st quartile (0

            15). After adjusting personal associated factors and family-associated factors such as parents’ physical status, parents’ mental health status, and parents’ marital status, college students with MHKQ scoring in the 3rd quartile (17

            0.99) and the 4th quartile (18

            0.97) were associated with a lowered risk of depressive symptoms, compared with MHKQ scoring in the 1st quartile (0

            6 (Initial) conclusions and areas for future research

            AI has attracted significant antitrust interest, raising the question of whether the competition regimes as they stand are ready to address potentially anti-competitive outcomes arising from AI decisions. Although many issues arising with AI elide with the antitrust debate around Big data, AI and the use of algorithms raises its own rather specific issues.

            The AI antitrust scholarship makes a bold claim that AI is an enabler of tacit collusion and could increase the scope for anti-competitive outcomes at even lower levels of concentration than associated with antitrust orthodoxy. However, even the brief examination of these claims in this article has unearthed alternative hypotheses which need to be fully tested before the theory can be incorporated in policy and legal environments without running the risk of being counter-productive.

            • Analysis of the effects of algorithms on incentives for tacit collusion and their destabilizing effects including in markets which are not already prone to tacit collusion. In particular, this involves understanding how robust the predictions in the AI literature are to their assumptions (e.g. algorithmic heterogeneity, larger number of sellers etc).
            • Whether there might be alternative outcomes which present competition or other concerns but which are not caught within traditional antitrust paradigms (e.g. data capture, data extraction and co-opetition between super-platforms and applications developers).
            • Understanding rational and harmful price transparency and whether and when particular consumer outcomes are an appropriate case for antitrust intervention. This accepts that consumers make bad decisions even in competitive markets and that instances of consumers making bad decisions caused by algorithmic pricing may not be an appropriate case for antitrust intervention.
            • Understanding countervailing AI strategies by buyers under a range of assumptions, including across B2C and B2B markets.
            • Understanding the appropriate boundaries of liability and the circumstances in which an algorithm may be traced back to its owners and the extent to which those owners should be subject to (vicarious) antitrust responsibilities and enforcement.
            • Understanding the main goals of antitrust which are impacted by AI. The current resistance on the part of regulators in Europe and the United States to regulate wealth transfers between AI sellers and buyers could place limits on the application of antitrust to consumer exploitation such as through data extraction. Ezrachi and Stucke have, however, presented an additional gloss in the idea that virtual competition increases the ‘dead weight loss by increasing distrust’. 44 Further examination is needed as to whether presenting the social costs of algorithms within a paradigm of (mis)trust provides an appropriate analytical construct that is capable of real-world application so as to legitimize antitrust interventions within a coherent welfare-based model.
            • Where personal data is shared with another market participant, the extent to which such data sharing would involve sharing of competitively sensitive information with a competitor and how such sharing may be compatible with antitrust law. 45

            Ezrachi A. & Stucke M. , Virtual Competition: The Promise and Perils of the Algorithm-Driven Economy , ( Harvard University Press , 2016 ).

            Virtual Competition: The Promise and Perils of the Algorithm-Driven Economy 2016

            Tacit collusion is a form of collusion typically seen in an oligopolistic market structure, where competing firms providing a good do not explicitly collude on any feature (such as price, quantity, or product characteristics), but rather, observe and imitate each other's actions in a way that is mutually beneficial to both sides.

            Co-opetition involves collaboration between competitors, in the hope of mutually beneficial results.

            See, further, Artificial Intelligence, Emerging Opportunities, Challenges, and Implications, Highlights of a Forum Convened by the Comptroller General of the United States (March 2018, GAO-18-142SP), available at:

            Russel S. & Norvig P. , Artificial Intelligence: A Modern Approach , ( Pearson , 2010 ).

            Artificial Intelligence: A Modern Approach 2010 3

            Nilsson N. , The Quest for Artificial Intelligence: A History of Idea and Achievement , ( Cambridge University Press , 2009 ).

            The Quest for Artificial Intelligence: A History of Idea and Achievement 2009

            Cambridge University Press

            Regulation (EU) 2016/679 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation) O.J. 2016 L 119/1 (‘GDPR’).

            The personal data requiring greater protection under the GDPR may be grouped into the following broad categories: Special categories of data. Special categories of data are, for the purposes of processing under the GDPR: personal data revealing racial or ethnic origin, political opinions, religious or philosophical beliefs, or trade union membership, and the processing of genetic data, biometric data for the purposes of uniquely identifying a natural person, data concerning health or data concerning a natural person's sex life or sexual orientation (Article 9(1), GDPR). Criminal conviction and offence data. Processing of personal data relating to criminal convictions and offences or related security measures based on Article 6(1) shall be carried out only under the control of official authority or when the processing is authorised by Union or Member State law providing for appropriate safeguards for the rights and freedoms of data subjects. Any comprehensive register of criminal convictions shall be kept only under the control of official authority (Article 10, GDPR).

            S. Mehra, ‘Antitrust and the Robo-Seller: Competition in the Time of Algorithms’ ( 2006 ) 100 Minnesota Law Review 1323 – 75 .

            Data perturbation is a data security technique that modifies the database to preserve the privacy and confidentiality of the data.

            Data masking is a method of creating a structurally similar but inauthentic version of an organization's data that can be used for purposes such as user training or software testing.

            Communication from the European Commission – Guidelines on the applicability of Article 101 of the Treaty on the Functioning of the European Union to horizontal co-operation agreements, O.J. 2011 C11/1 (‘Horizontal Cooperation Guidelines’).

            JJB Sports plc v Office of Fair Trading Allsports Limited v Office of Fair Trading [2006] EWCA Civ 1318, para 141.

            The European Data Protection Supervisor (‘EDPS’) is the EU's independent data protection authority.

            Opinion 8/2016, EDPS Opinion on coherent enforcement of fundamental rights in the age of big data (23 September 2016) p 6, available at:

            See, further, Case C-82/01 Aeroports de Paris v Commission EU:C:2002:61 and section C (priced-based exclusionary conduct), Communication from the European Commission: Guidance on its enforcement priorities in applying Article 82 of the EC Treaty to abusive exclusionary conduct by dominant undertakings, O.J. 2009 C45/7 (Article 82 ‘Priority Guidance’).

            See, e.g., Aeroports de Paris (fn 16).

            During periods of excessive demand or scarce supply, when there are far more riders than drivers, Uber increases its normal fares with a multiplier whose value depends on the scarcity of available drivers. See, further: ‘Surge pricing: How it works and how to avoid it’ (BBC, 15 January 2018), available at:

            CMA, Energy Market Investigation, Final Report (24 June 2016).

            Energy Market Investigation (fn 20), para 175.

            CMA, Consumer Protection: Enforcement Guidance (17 August 2016) (CM58) (‘Consumer Protection Enforcement Guidance’), para 2.2.

            European Commission, Final report on the E-Commerce Sector Inquiry (COM(2017) 229 final). Case page available at:

            CMA, Market Study on Digital Comparison Tools, Final Report (26 September 2017).

            CMA, Private motor insurance market investigation, Final Report (24 September 2014).

            Directive 2005/29/EC concerning unfair business-to-consumer commercial practices in the internal market, O.J. 2005 L149/22.

            These principles were developed in conjunction with the 2017 Asilomar conference. Further information can be found at

            European Parliament resolution of 16 February 2017 with recommendations to the Commission on Civil Law Rules on Robotics (2015/2103(INL)), O.J. 2018 C 252/239.

            In September 2000 Amazon offered to sell a buyer a DVD for one price, but after the buyer deleted cookies that identified him as a regular Amazon customer, he was offered the same DVD for a substantially lower price. Amazon's CEO Jeff Bezos subsequently apologised for the differential pricing and promised that Amazon ‘never will test prices based on customer demographics’: see ‘Bezos calls Amazon experiment a mistake’ (28 September 2000), available at:

            Bundeskartellamt 18th Conference on Competition Berlin (16 March 2017).

            See, further, the European Commission's guidance to businesses on compliance with competition law, available on its website at

            European Commission , Competition Policy for the digital era , Final report ( April 2019 ). Available at: .

            See, further, Competition Policy for the digital era (fn 32), ch 3.

            Case 39740 Google Search (Shopping), European Commission decision of 27 June 2017 (C(2017) 4444 final). Case page available at:

            Case 50565-2 Digital piano and digital keyboard sector: anti-competitive practices (1 August 2019).

            ‘David Currie on the role of competition in stimulating innovation’, Speech given by CMA Chairman, David Currie, at the Concurrences Innovation Economics Conference, King's College London (3 February 2017), available at:

            CMA , Pricing algorithms, Economic working paper on the use of algorithms to facilitate collusion and personalised pricing ( 8 October 2018 ) (CMA94) (‘Pricing algorithms’), available at: .

            Pricing algorithms (fn 37), para 7.17.

            Pricing algorithms (fn 37), para 8.6.

            Pricing algorithms (fn 37), para 5.37.

            Department of Justice press release No 15-421, Former E-Commerce Executive Charged with Price Fixing in the Antitrust Division's First Online Marketplace Prosecution (6 April 2015).

            Case No 37 of 2018, Order of 6 November 2018, available at:

            Section 3 of the Indian Competition Act 2002 was informed by and is the practical equivalent under Indian competition law to Article 101 TFEU and Chapter I of the UK Competition Act 1998.

            Ezrachi and Stucke (fn 1), p 242.

            A current debate is whether there is an enforcement gap in mergers in relation to lost potential competition in such situations and in Big data scenarios more generally. This issue of whether AI in the hands of a limited number of data controllers is adequately addressed under merger control was not explored in this article.

            Watch the video: Επαλήθευση Διαίρεσης παράδειγμα με τέλεια και ατελή διαίρεση. Proper Education (December 2021).