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.
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Here are the solutions for question 19: Question 19a) To find the value of 'w', we use the formula for the average (mean) score: Average = (Mark × No. of pupils) (No. of pupils) Step 1: Calculate the sum of (Mark × No. of pupils). (Mark × No. of pupils) = (45 × 2) + (60 × w) + (50 × 4) + (70 × 1) = 90 + 60w + 200 + 70 = 360 + 60w Step 2: Calculate the total number of pupils. (No. of pupils) = 2 + w + 4 + 1 = 7 + w Step 3: Substitute the given average score (54) and the sums into the average formula. 54 = (360 + 60w)/(7 + w) Step 4: Solve for w. 54(7 + w) = 360 + 60w 378 + 54w = 360 + 60w 378 - 360 = 60w - 54w 18 = 6w w = (18)/(6) w = 3 Question 19b) To find the median score, we first need to update the table with the value of w=3. | Marks | No. of pupils | |-------|---------------| | 45 | 2 | | 60 | 3 | | 50 | 4 | | 70 | 1 | Step 1: Find the total number of pupils (N). N = 2 + 3 + 4 + 1 = 10 Step 2: Order the marks from lowest to highest and list the scores. The ordered scores are: 45, 45 (from 2 pupils scoring 45) 50, 50, 50, 50 (from 4 pupils scoring 50) 60, 60, 60 (from 3 pupils scoring 60) 70 (from 1 pupil scoring 70) Step 3: Determine the position of the median. Since N = 10 (an even number), the median is the average of the two middle values. These are the (N)/(2)-th and ((N)/(2) + 1)-th values. The positions are the (10)/(2) = 5-th score and the ((10)/(2) + 1) = 6-th score. From the ordered list: The 5th score is 50. The 6th score is 50. Step 4: Calculate the median. Median = 5th score + 6th score2 Median = (50 + 50)/(2) Median = (100)/(2) Median = 50