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Calculate the value of M and identify the minimum value of the data. Also, calculate the missing value C.

Mathematics

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Here's the solution to Question 4:

4.1.1 Calculate the value of M if the mean shoe size is 7.

  1. List all the given shoe sizes and set up the mean formula.

    The shoe sizes are 6,9,8,7,6,M,6,8,7,96, 9, 8, 7, 6, M, 6, 8, 7, 9. There are 10 data points.
    The formula for the mean is:

    Mean=Sum of all data pointsNumber of data points\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} 7=6+9+8+7+6+M+6+8+7+9107 = \frac{6 + 9 + 8 + 7 + 6 + M + 6 + 8 + 7 + 9}{10}

  2. Sum the known values and solve for M.

    7=66+M107 = \frac{66 + M}{10}

    Multiply both sides by 10:

    7×10=66+M7 \times 10 = 66 + M 70=66+M70 = 66 + M

    Subtract 66 from both sides:

    M=70−66M = 70 - 66 M=4M = \boxed{\text{4}}

    4.1.2 Identify the minimum value of the data.

  3. Substitute the value of M into the data set.

    The data set becomes: $6, 9, 8, 7, 6, 4, 6, 8, 7, 9$.

  4. Identify the smallest value in the data set.

    Arranging the data in ascending order: 4,6,6,6,7,7,8,8,9,94, 6, 6, 6, 7, 7, 8, 8, 9, 9.
    The minimum value is 4\boxed{\text{4}}.

    4.2.1 Calculate the missing value C.

  5. Identify C as the Meal Allowance and use the Total Earnings formula.

    From the payslip, C is the Meal Allowance.
    Total Earnings = Basic Salary + Incentive Pay + House Allowance + Meal Allowance (C)

    R14 800.14=R12 000+(16.67% of R12 000)+R500+C\text{R}14\,800.14 = \text{R}12\,000 + (16.67\% \text{ of R}12\,000) + \text{R}500 + C

  6. Calculate the Incentive Pay.

    Incentive Pay=0.1667×R12 000=R2 000.40\text{Incentive Pay} = 0.1667 \times \text{R}12\,000 = \text{R}2\,000.40

    (Assuming 16.67% is approximately 1/61/6, which gives R2 000. We will use R2 000.40 for precision based on the given percentage.)

  7. Substitute the values and solve for C.

    R14 800.14=R12 000+R2 000.40+R500+C\text{R}14\,800.14 = \text{R}12\,000 + \text{R}2\,000.40 + \text{R}500 + C R14 800.14=R14 500.40+C\text{R}14\,800.14 = \text{R}14\,500.40 + C C=R14 800.14−R14 500.40C = \text{R}14\,800.14 - \text{R}14\,500.40 C=R299.74C = \boxed{\text{R}299.74}

    4.2.2 Determine the missing value D, the Net Pay.

  8. Calculate the Total Deductions.

    Total Deductions = Provident Fund + Tax + Loan

    Total Deductions=R1 500+R700+R600\text{Total Deductions} = \text{R}1\,500 + \text{R}700 + \text{R}600 Total Deductions=R2 800\text{Total Deductions} = \text{R}2\,800

  9. Calculate the Net Pay (D) using Total Earnings and Total Deductions.

    Net Pay = Total Earnings - Total Deductions

    D=R14 800.14−R2 800D = \text{R}14\,800.14 - \text{R}2\,800 D=R12 000.14D = \boxed{\text{R}12\,000.14}
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