Question : The average of a certain set of 'b' numbers is 'a', and the average of a certain set of 'a' numbers is 'b'. What is the average of all the numbers taken together?
Option 1: 2b ÷ (a+b)
Option 2: 2ab ÷ (a+b)
Option 3: (a+b) ÷ 2ab
Option 4: 2a ÷ (a+b)
Correct Answer: 2ab ÷ (a+b)
Solution :
Average of the set of b numbers = a
Average of the set of a numbers = b
Average = $\frac{\text{Total sum of value}}{\text{Number of values}}$
The sum of the set of b numbers = Average × Total values
The sum of the set of b numbers = a × b = ab
The sum of the set of a numbers = b × a = ba
Total Sum of all numbers = ab + ba = 2ab
Total values = a + b
Average of all numbers = $\frac{2\text{ab}}{\text{a}+\text{b}}$
Hence, the average of all the numbers taken together is $\frac{2\text{ab}}{\text{a}+\text{b}}$.
Hence, the correct answer is 2ab ÷ (a+b).
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