Multiple choice

The frequency distribution of marks obtained by $28$ students in a test carrying $40$ marks is given below Marks 0-10 10-20 20-30 30-40 Number of students 6 $x$ $y$ 6 If the mean of the above data is $20$, then the difference between $x$ and $y$ is

  1. $3$
  2. $2$
  3. $1$
  4. $0$
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D Correct answer
Explanation

Midpoints: 5, 15, 25, 35. Mean = (5*6 + 15*x + 25*y + 35*6) / 28 = 20. 30 + 15x + 25y + 210 = 560. 15x + 25y = 320. Divide by 5: 3x + 5y = 64. Also 6+x+y+6 = 28, so x+y = 16. Solve: 3x + 5(16-x) = 64 -> 3x + 80 - 5x = 64 -> -2x = -16 -> x=8. Then y=8. Difference x-y = 0.

AI explanation

The total number of students is 6 plus x plus y plus 6, which simplifies to x plus y plus 12, and since there are 28 students, x plus y equals 16. Using the step-deviation method for the mean, the sum of the midpoints multiplied by their frequencies is (5 times 6) plus (15 times x) plus (25 times y) plus (35 times 6), equaling 30 plus 15x plus 25y plus 210. Because the overall mean is 20 across 28 students, the total sum must be 560, so 240 plus 15x plus 25y equals 560; dividing by 5 gives 3x plus 5y equals 64. Solving the system where x plus y equals 16 and 3x plus 5y equals 64 results in x equaling 8 and y equaling 8, making their difference 0.

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