This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Calculate the midpoint (x) for each mark range. The midpoint of a class interval is calculated as lower limit + upper limit2. | Marks | Frequency (f) | Midpoint (x) | fx | | :-------- | :-------------- | :------------- | :---------- | | 11-15 | p | (11+15)/(2) = 13 | 13p | | 16-20 | 4 | (16+20)/(2) = 18 | 4 × 18 = 72 | | 21-25 | 11 | (21+25)/(2) = 23 | 11 × 23 = 253 | | 26-30 | k | (26+30)/(2) = 28 | 28k | | 31-35 | 6 | (31+35)/(2) = 33 | 6 × 33 = 198 | | 36-40 | 1 | (36+40)/(2) = 38 | 1 × 38 = 38 | Step 2: Form an equation using the total number of students. The total number of students is given as 40. The sum of the frequencies must equal 40. p + 4 + 11 + k + 6 + 1 = 40 p + k + 22 = 40 p + k = 40 - 22 p + k = 18 (Equation 1) Step 3: Form an equation using the mean mark. The mean mark is given as 24.75. The formula for the mean of a grouped frequency distribution is Mean = ( fx)/( f). First, calculate fx: fx = 13p + 72 + 253 + 28k + 198 + 38 fx = 13p + 28k + 561 Now, substitute this into the mean formula along with f = 40 and Mean = 24.75: 24.75 = (13p + 28k + 561)/(40) Multiply both sides by 40: 24.75 × 40 = 13p + 28k + 561 990 = 13p + 28k + 561 13p + 28k = 990 - 561 13p + 28k = 429 (Equation 2) Step 4: Solve the system of linear equations for p and k. From Equation 1, express p in terms of k: p = 18 - k Substitute this expression for p into Equation 2: 13(18 - k) + 28k = 429 234 - 13k + 28k = 429 234 + 15k = 429 15k = 429 - 234 15k = 195 k = (195)/(15) k = 13 Now substitute the value of k back into Equation 1 to find p: p + 13 = 18 p = 18 - 13 p = 5 The values are p=5 and k=13. The final answer is p=5, k=13. That's 4 down. 1 left today — send the next one.