Multiple choice

There are three classes X, Y and Z. Average score of class X is 83. Average score of class Y is 76. Average score of class Z is 85. Average score of classes X and Y is 79, average score of classes Y and Z is 81. Find the average score of all three classes.

  1. 81

  2. 81.5

  3. 82

  4. 84
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B Correct answer
Explanation

Let nX, nY, nZ be the number of students. Average of X and Y: (83nX + 76nY)/(nX + nY) = 79 => 4nX = 3nY => nY = 4/3 nX. Average of Y and Z: (76nY + 85nZ)/(nY + nZ) = 81 => 5nY = 4nZ => nZ = 5/4 nY = 5/4 * 4/3 nX = 5/3 nX. Total average = (83nX + 76(4/3 nX) + 85(5/3 nX)) / (nX + 4/3 nX + 5/3 nX) = (83 + 304/3 + 425/3) / (1 + 4/3 + 5/3) = (249 + 304 + 425)/3 / (12/3) = 978 / 12 = 81.5.

AI explanation

Let the number of students in classes X, Y, and Z be x, y, and z respectively. Using the given combined averages, we form the equations 83x plus 76y equals 79 times x plus y, and 76y plus 85z equals 81 times y plus z. Solving these equations gives the ratios x:y as 3:4 and y:z as 4:5, making the ratio x:y:z equal to 3:4:5. Assuming 3, 4, and 5 students for the classes, the total score is 249 plus 304 plus 425, which equals 978, and dividing this by the total of 12 students gives an overall average of 81.5.

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