Reasonining

• Mean , mean is when you add all the values and then divide by the number of values .
• If we assume that one friend paid an extra £1, this would make up the difference. However, if two friends had paid an extra £1 each, it still wouldn’t be enough as the total would only be £27. Therefore, at least three people need to pay more than £6 each.
• The total amount spent on tickets = 5 x £6 = £30. The total amount collected from their pocket money = £21 (£3 + £8 + £6 + £4). So there is a shortfall of £9 which needs to be found elsewhere.
• So now we have two equations with two unknowns ($x$ and $y$). Let's solve these simultaneously using substitution or elimination.
• We can see that the person who paid £3 must have given some change back so let's call this value 'x'. We know that the other four friends gave no change back because they all paid exactly what was needed. This means that the total amount of change given out was £x. Since the total amount of money spent was £30, we can write down the equation:
• To find out how many students are younger than 15, we can subtract the ages of those who are older or equal to 15 from the total sum of their ages (which equals twice the product of the average age and the number of students). The remaining amount represents the sum of the ages of those who are younger than 15. We can now use the formula for finding the average of a set of numbers to calculate the average age of these students. Finally, we multiply the average with the number of students who are younger than 15 to get the exact number of such students.
• the median is when a number is in the middle of a date set (let’s say line of numbers).
• range is the distance between the highest and lowest values in a dataset.
• Mode is the most frequent Number that comes up in a set of data.
• How is each triangular number formed?

  Each triangular number is formed by adding the next consecutive integer to the previous triangular number.
• What are triangular numbers?

  Triangular numbers are a sequence of numbers that can be represented by a triangular arrangement of dots.
• What is the pattern of triangular numbers?

  • 1st: 1 = 1
  • 2nd: 1 + 2 = 3
  • 3rd: 1 + 2 + 3 = 6
  • 4th: 1 + 2 + 3 + 4 = 10
  • 5th: 1 + 2 + 3 + 4 + 5 = 15
• What is the formula for calculating the nth triangular number?

  The formula for the nth triangular number is $T_n =$$\frac{n(n+1)}{2}$.
• How do you calculate the 5th triangular number using the formula?

  To find the 5th triangular number: $T_5 =$$\frac{5(5+1)}{2} =$$15$.
• What is the significance of the formula for triangular numbers?

  The formula allows for quick calculation of any triangular number without summing all previous numbers.
• What are some properties of triangular numbers?

  1. Each triangular number is the sum of consecutive integers from 1 to n.
  2. The sum of two consecutive triangular numbers equals a square number.
  3. Triangular numbers alternate between odd and even.
  4. Every triangular number is divisible by the number of rows in its triangle representation.
  5. Every other triangular number is a Fibonacci number.
• How do triangular numbers relate to square numbers?

  The sum of two consecutive triangular numbers always equals a square number.
• What is the 7th triangular number and its properties?

  The 7th triangular number is 28, which is even and divisible by 7.
• What is the 5th triangular number and its properties?

  The 5th triangular number is 15, which is odd and divisible by 5.
• What are some real-world applications of triangular numbers?

  1. Combinatorics: Represent ways to choose 2 items from n items.
  2. Data structures: Used in algorithms for efficient data storage.
  3. Game theory: Applied in strategies and probability calculations.
  4. Physical arrangements: Used in setups like bowling pins.
  5. Project management: Models cumulative task completion.
• How many matches will be played in a round-robin tournament with 6 teams?

  There will be 15 matches played, calculated using the formula for triangular numbers: $T_5 =$$\frac{5(5+1)}{2} =$$15$.
• What are triangular numbers used for in combinatorics?

  They represent the number of ways to choose 2 items from n items.
• How are triangular numbers applied in data structures?

  They are used in certain algorithms for efficient data storage and retrieval.
• In what context are triangular numbers used in game theory?

  They are applied in some game strategies and probability calculations.
• Give an example of a physical arrangement that uses triangular numbers.

  Bowling pin setups.
• How can triangular numbers be used in project management?

  They can model cumulative task completion in certain project schedules.