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Solve the equation 1-4x / 3 = (3x−5)/2 and find the mode, median, and mean of the scores from the table.

Mathematics

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3 parts
a

1\text{1}

b

2\text{2}

c

2.3\text{2.3}

1\text{1}

Step-by-step solution

10 steps

Here are the solutions to the problems.

1. A.
The table shows the scores (m) and their frequencies (f):

Scores (m)Freq. (f)
18
24
34
42
52

The total number of students is the sum of frequencies:

N=8+4+4+2+2=20N = 8 + 4 + 4 + 2 + 2 = 20

Part a· 1 step

Find the mode

  1. Identify the score with the highest frequency.

    The highest frequency is 8, which corresponds to a score of 1.
    The mode is the value that appears most frequently in a data set.

    Mode=1\text{Mode} = \boxed{\text{1}}

Part b

2\text{2}

Find the median

  1. Determine the position of the median.

    Since the total number of data points N=20N=20 (an even number), the median is the average of the (N2)\left(\frac{N}{2}\right)-th and (N2+1)\left(\frac{N}{2}+1\right)-th values.
    The positions are the 202=10\frac{20}{2} = 10-th and 202+1=11\frac{20}{2}+1 = 11-th values.

  2. Find the values at these positions using cumulative frequencies.

    • Scores of 1 account for the 1st to 8th values.
    • Scores of 2 account for the 9th to 12th values (8+4=128+4=12). Both the 10th and 11th values are 2.

  3. Calculate the median.

    Median=10th value+11th value2=2+22=42=2\text{Median} = \frac{10\text{th value} + 11\text{th value}}{2} = \frac{2+2}{2} = \frac{4}{2} = \boxed{\text{2}}

Part c

2.3\text{2.3}

Find the mean of the scores

  1. Calculate the sum of (score ×\times frequency).

    ∑(m×f)=(1×8)+(2×4)+(3×4)+(4×2)+(5×2)\sum (m \times f) = (1 \times 8) + (2 \times 4) + (3 \times 4) + (4 \times 2) + (5 \times 2) ∑(m×f)=8+8+12+8+10=46\sum (m \times f) = 8 + 8 + 12 + 8 + 10 = 46

  2. Calculate the mean using the formula Mean=∑(m×f)∑f\text{Mean} = \frac{\sum (m \times f)}{\sum f}.

    Mean=4620=2.3\text{Mean} = \frac{46}{20} = \boxed{\text{2.3}}

    2. A.
    Solve the equation 3x−51−4x=23\frac{3x-5}{1-4x} = \frac{2}{3}.

  3. Cross-multiply to eliminate the denominators.

    3(3x−5)=2(1−4x)3(3x-5) = 2(1-4x)

  4. Distribute the numbers on both sides of the equation.

    9x−15=2−8x9x - 15 = 2 - 8x

  5. Collect all terms with xx on one side and constant terms on the other side.

    9x+8x=2+159x + 8x = 2 + 15 17x=1717x = 17

  6. Isolate xx by dividing both sides by 17.

    x=1717x = \frac{17}{17} x=1x = \boxed{\text{1}}
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