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Calculate how much more Mr Mabuza will pay per month for the return trips after the increase. Also, determine the mode of the learners' masses data set and explain why it's called the mode.

Mathematics

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here are the solutions to the questions:

Part 2.2· 3 steps

2 Calculate how much more Mr Mabuza will pay per month for the return trips after the increase.

  1. Determine the increase in fare per single trip.

    From the previous question (2.2.1), the original fare was R15.00 and the new fare is R18.00.

    Increase per single trip=New fare−Original fare\text{Increase per single trip} = \text{New fare} - \text{Original fare} Increase per single trip=R 18.00−R 15.00=R 3.00\text{Increase per single trip} = \text{R } 18.00 - \text{R } 15.00 = \text{R } 3.00

  2. Calculate the increase in fare per return trip.

    A return trip consists of two single trips.

    Increase per return trip=2×R 3.00=R 6.00\text{Increase per return trip} = 2 \times \text{R } 3.00 = \text{R } 6.00

  3. Calculate the total additional cost per month for 20 return trips.

    Total additional cost=Number of return trips×Increase per return trip\text{Total additional cost} = \text{Number of return trips} \times \text{Increase per return trip} Total additional cost=20×R 6.00=R 120.00\text{Total additional cost} = 20 \times \text{R } 6.00 = \text{R } 120.00

    Mr Mabuza will pay R 120.00 more per month.

Part 3

The given data set of learners' masses (in kg) is: 48; 51; 59; 63; 69; 36; 54; 60; 63; 71; 68; 48; 37; 48; 53; 50.
First, let's order the data set for easier calculations:
36, 37, 48, 48, 48, 50, 51, 53, 54, 59, 60, 63, 63, 68, 69, 71.
There are 16 data points.

Part 3.1· 2 steps

Determine the mode of the data set and explain why that value is referred to as a mode.

  1. Identify the frequency of each value.

    • 48 appears 3 times.
    • 63 appears 2 times.
    • All other values appear once.

  2. Determine the mode.

    The mode is the value that appears most frequently in the data set.
    The mode is 48 kg. This value is referred to as the mode because it is the mass that occurs most often among the learners.

Part 3.2

Define the term median in the context above.

The median is the middle value in a data set when the values are arranged in numerical order. If there is an even number of data points, the median is the average of the two middle values.

Part 3.3· 3 steps

Determine the median value.

  1. The data set is already ordered

    36, 37, 48, 48, 48, 50, 51, 53, 54, 59, 60, 63, 63, 68, 69, 71.
    There are 16 data points, so the median is the average of the 8th and 9th values.

  2. Identify the 8th and 9th values.

    The 8th value is 53.
    The 9th value is 54.

  3. Calculate the median.

    Median=53+542\text{Median} = \frac{53 + 54}{2} Median=1072\text{Median} = \frac{107}{2} Median=53.5 kg\text{Median} = 53.5 \text{ kg}

    The median value is 53.5 kg.

Part 3.4· 2 steps

Lebohang states that the range of the above data set is 35 kg, use calculations to whether the statement is valid or not.

  1. Identify the maximum and minimum values from the ordered data set.

    Maximum value = 71 kg
    Minimum value = 36 kg

  2. Calculate the range.

    Range=Maximum value−Minimum value\text{Range} = \text{Maximum value} - \text{Minimum value} Range=71 kg−36 kg\text{Range} = 71 \text{ kg} - 36 \text{ kg} Range=35 kg\text{Range} = 35 \text{ kg}

    Lebohang's statement is valid because the calculated range is 35 kg.

Part 3.5· 2 steps

Calculate the average mass of the data set.

  1. Sum all the masses.

    Sum=36+37+48+48+48+50+51+53+54+59+60+63+63+68+69+71\text{Sum} = 36 + 37 + 48 + 48 + 48 + 50 + 51 + 53 + 54 + 59 + 60 + 63 + 63 + 68 + 69 + 71 Sum=878 kg\text{Sum} = 878 \text{ kg}

  2. Calculate the average (mean) mass.

    There are 16 learners.

    Average mass=Sum of massesNumber of learners\text{Average mass} = \frac{\text{Sum of masses}}{\text{Number of learners}} Average mass=878 kg16\text{Average mass} = \frac{878 \text{ kg}}{16} Average mass=54.875 kg\text{Average mass} = 54.875 \text{ kg}

    The average mass of the data set is 54.875 kg.

Part 3.6

Is the above data discrete or continuous? Explain your choice.

The data is continuous. Mass is a measurement that can take any value within a given range, including decimals and fractions, rather than being restricted to distinct, separate values.

Part 3.7· 2 steps

Determine the probability (as a decimal) of having a learner that weighs less than 50 kg.

  1. Count the number of learners who weigh less than 50 kg from the ordered data set.

    The masses less than 50 kg are: 36, 37, 48, 48, 48.
    There are 5 learners who weigh less than 50 kg.

  2. Calculate the probability.

    Total number of learners = 16.

    Probability=Number of learners weighing less than 50 kgTotal number of learners\text{Probability} = \frac{\text{Number of learners weighing less than 50 kg}}{\text{Total number of learners}} Probability=516\text{Probability} = \frac{5}{16} Probability=0.3125\text{Probability} = 0.3125

    The probability is 0.3125.

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