We read your photo as:

A frequency distribution table shows marks obtained by students. The first quartile (Q1) is 35. Find the first quartile class and the value of x.

Mathematics

ScanSolve AI Answer

Answers

4 parts
a

20-40\text{20-40}

b
4
c
51.43
d
7:1

Step-by-step solution

12 steps

Here are the solutions to the questions based on the provided data:

The given frequency distribution table is:

Marks ObtainedNo. of Students (f)
0-202
20-40x
40-608
60-805
80-1001

The first quartile (Q1Q_1) of the given data is 35.

First, we construct the cumulative frequency (cf) table:

Marks ObtainedNo. of Students (f)Cumulative Frequency (cf)
0-2022
20-40x2+x2 + x
40-6082+x+8=10+x2 + x + 8 = 10 + x
60-80510+x+5=15+x10 + x + 5 = 15 + x
80-100115+x+1=16+x15 + x + 1 = 16 + x

The total number of students is N=16+xN = 16 + x.

Part a· 1 step

Find the first quartile class.

  1. Identify the class interval where the first quartile value (35) falls.

    Since 20<35<4020 < 35 < 40, the first quartile class is 20-40.

    The first quartile class is 20-40\boxed{\text{20-40}}.

Part b4

Find the value of x.

  1. Use the formula for the first quartile (Q1Q_1)

    Q1=L+N4−cff×hQ_1 = L + \frac{\frac{N}{4} - cf}{f} \times h

    From part (a), the first quartile class is 20-40.
    L=20L = 20 (lower boundary of the quartile class)
    f=xf = x (frequency of the quartile class)
    cf=2cf = 2 (cumulative frequency of the class preceding the quartile class)
    h=40−20=20h = 40 - 20 = 20 (class width)
    Q1=35Q_1 = 35 (given)
    N=16+xN = 16 + x (total number of students)

  2. Substitute the values into the formula

    35=20+16+x4−2x×2035 = 20 + \frac{\frac{16+x}{4} - 2}{x} \times 20
  3. Solve for xx

    35−20=16+x4−2x×2035 - 20 = \frac{\frac{16+x}{4} - 2}{x} \times 20 15=16+x−84x×2015 = \frac{\frac{16+x-8}{4}}{x} \times 20 15=8+x4x×2015 = \frac{\frac{8+x}{4}}{x} \times 20 15=8+x4x×2015 = \frac{8+x}{4x} \times 20 15=8+xx×515 = \frac{8+x}{x} \times 5

    Divide both sides by 5:

    3=8+xx3 = \frac{8+x}{x} 3x=8+x3x = 8+x 3x−x=83x - x = 8 2x=82x = 8 x=82x = \frac{8}{2} x=4x = 4

    The value of xx is 4\boxed{4}.

Part c51.43

If modal class is 40-60 then what is the value of mode?

  1. Update the frequency table with x=4x=4

    Marks ObtainedNo. of Students (f)
    0-202
    20-404
    40-608
    60-805
    80-1001
  2. Use the formula for the mode for grouped data

    Mode=L+fm−f12fm−f1−f2×hMode = L + \frac{f_m - f_1}{2f_m - f_1 - f_2} \times h

    The modal class is given as 40-60.
    L=40L = 40 (lower boundary of the modal class)
    fm=8f_m = 8 (frequency of the modal class)
    f1=4f_1 = 4 (frequency of the class preceding the modal class, 20-40)
    f2=5f_2 = 5 (frequency of the class succeeding the modal class, 60-80)
    h=60−40=20h = 60 - 40 = 20 (class width)

  3. Substitute the values into the formula

    Mode=40+8−42(8)−4−5×20Mode = 40 + \frac{8 - 4}{2(8) - 4 - 5} \times 20 Mode=40+416−4−5×20Mode = 40 + \frac{4}{16 - 4 - 5} \times 20 Mode=40+412−5×20Mode = 40 + \frac{4}{12 - 5} \times 20 Mode=40+47×20Mode = 40 + \frac{4}{7} \times 20 Mode=40+807Mode = 40 + \frac{80}{7} Mode=40+11.42857...Mode = 40 + 11.42857... Mode≈51.43Mode \approx 51.43

    The value of the mode is 51.43\boxed{51.43}.

Part d7:1

Find the ratio of students who are above and below the first quartile class.

  1. Identify the first quartile class.

    From part (a), the first quartile class is 20-40.

  2. Calculate the number of students below the first quartile class.

    Students below the 20-40 class are those in the 0-20 class.
    Number of students below the first quartile class = 2.

  3. Calculate the number of students above the first quartile class.

    Students above the 20-40 class are those in the 40-60, 60-80, and 80-100 classes.
    Number of students above the first quartile class = 8+5+1=148 + 5 + 1 = 14.

  4. Find the ratio of students above to students below the first quartile class.

    Ratio = (Number of students above) : (Number of students below)
    Ratio = $14 : 2$

  5. Simplify the ratio.

    Divide both sides by 2:
    Ratio = 7:17 : 1

    The ratio of students who are above and below the first quartile class is 7:1\boxed{7:1}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question