In a class, the average score of 8 boys is 56 and the average score of 4 girls is 80. What is the average score of all 12 student…

Question

In a class, the average score of 8 boys is 56 and the average score of 4 girls is 80. What is the average score of all 12 students?

1. A. 62
2. B. 66
3. C. 68
4. D. 64 (Correct answer)

Correct Answer

Option D — 64

Detailed Solution & Explanation

The correct answer is 64.

Shortcut Trick

• Use weighted average: the boys outnumber the girls 2 : 1, so the combined mean sits twice as close to 56 as to 80.
• The gap is $80 - 56 = 24$; splitting it in the ratio $1 : 2$ from the boys' side gives $56 + \dfrac{24}{3} = 56 + 8 = \textbf{64}$.

Alternate Method

• Total score of boys $= 8 \times 56 = 448$
• Total score of girls $= 4 \times 80 = 320$
• Combined average $= \dfrac{448 + 320}{12} = \dfrac{768}{12} = \textbf{64}$

Additional Information

• The combined average is never the simple average of the two means unless the groups are equal in size. Here $\dfrac{56 + 80}{2} = 68$ — which is offered as a distractor precisely to catch that error.
• The alligation rule states that the distances from the combined mean to each group mean are in the inverse ratio of the group sizes: $8 : 4$ gives distances in the ratio $1 : 2$, i.e. 8 below 80's side and 16 above 56's.
• Weighted averaging generalises to any number of groups as $\dfrac{\sum n_i \bar{x}_i}{\sum n_i}$.