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Form a matrix of four by four to represent the information from four grade nine streams' mathematics contest results.

Mathematics

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*27. During a mathematics contest the following results were realised from four grade nine streams:

StreamNorthSouthEastWest
Exceeding Expectations91086
Meeting Expectations42914
Approaching Expectations111282
Below Expectations2314
Form a matrix of four by four to represent the information.*
  1. Identify the rows and columns from the table.

    Rows correspond to the expectation categories: Exceeding, Meeting, Approaching, Below.
    Columns correspond to the streams: North, South, East, West.

  2. Arrange the data into a 4×44 \times 4 matrix.

    (91086429141112822314)\begin{pmatrix} 9 & 10 & 8 & 6 \\ 4 & 2 & 9 & 14 \\ 11 & 12 & 8 & 2 \\ 2 & 3 & 1 & 4 \end{pmatrix}

    The matrix representing the information is (91086429141112822314)\boxed{\begin{pmatrix} 9 & 10 & 8 & 6 \\ 4 & 2 & 9 & 14 \\ 11 & 12 & 8 & 2 \\ 2 & 3 & 1 & 4 \end{pmatrix}}.

    28. Prepare a matrix of your choice of integers between -2 and 6 of order two by two.

  3. Understand the requirements.

    The matrix must be of order 2×22 \times 2, meaning it has 2 rows and 2 columns.
    The elements must be integers between -2 and 6 (inclusive). This means the integers can be -2, -1, 0, 1, 2, 3, 4, 5, 6.

  4. Choose four integers from the allowed range and arrange them into a 2×22 \times 2 matrix.

    For example, we can choose 1, 0, 5, and -1.

    A possible matrix is (105−1)\boxed{\begin{pmatrix} 1 & 0 \\ 5 & -1 \end{pmatrix}}.
    (Other valid answers are possible, as the choice is arbitrary within the given constraints.)

    29. Evaluate:

    (349460)−(−29−3−4−66)\begin{pmatrix} 3 & 4 \\ 9 & 4 \\ 6 & 0 \end{pmatrix} - \begin{pmatrix} -2 & 9 \\ -3 & -4 \\ -6 & 6 \end{pmatrix}
  5. Perform matrix subtraction by subtracting corresponding elements.

    (3−(−2)4−99−(−3)4−(−4)6−(−6)0−6)\begin{pmatrix} 3 - (-2) & 4 - 9 \\ 9 - (-3) & 4 - (-4) \\ 6 - (-6) & 0 - 6 \end{pmatrix}
  6. Simplify each element.

    (3+24−99+34+46+60−6)\begin{pmatrix} 3 + 2 & 4 - 9 \\ 9 + 3 & 4 + 4 \\ 6 + 6 & 0 - 6 \end{pmatrix} (5−512812−6)\begin{pmatrix} 5 & -5 \\ 12 & 8 \\ 12 & -6 \end{pmatrix}

    The evaluated matrix is (5−512812−6)\boxed{\begin{pmatrix} 5 & -5 \\ 12 & 8 \\ 12 & -6 \end{pmatrix}}.

    30. What is the order of

    (639340)\begin{pmatrix} 6 & 3 & 9 \\ 3 & 4 & 0 \end{pmatrix}
  7. Count the number of rows.

    There are 2 rows in the matrix.

  8. Count the number of columns.

    There are 3 columns in the matrix.

  9. State the order as (number of rows) ×\times (number of columns).

    The order of the matrix is 2×3\boxed{2 \times 3}.

    31. A girl is twice as old as a boy who is xx years old now. Their father is 25 years older than the girl. If the sum of their ages is 75, how old is the girl now?

  10. Define the ages of the boy, girl, and father in terms of xx.

    Boy's age =x= x years
    Girl's age =2×x=2x= 2 \times x = 2x years
    Father's age =Girl’s age+25=2x+25= \text{Girl's age} + 25 = 2x + 25 years

  11. Set up an equation for the sum of their ages.

    Sum of ages =Boy’s age+Girl’s age+Father’s age= \text{Boy's age} + \text{Girl's age} + \text{Father's age}

    75=x+2x+(2x+25)75 = x + 2x + (2x + 25)
  12. Solve the equation for xx.

    75=x+2x+2x+2575 = x + 2x + 2x + 25 75=5x+2575 = 5x + 25

    Subtract 25 from both sides:

    75−25=5x75 - 25 = 5x 50=5x50 = 5x

    Divide by 5:

    x=505x = \frac{50}{5} x=10x = 10
  13. Calculate the girl's age using the value of xx.

    Girl's age =2x=2×10=20= 2x = 2 \times 10 = 20 years

    The girl is 20 years old\boxed{20 \text{ years old}}.

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