The required average of all the numbers is 72.
What is the average of n numbers?
Let a₁, a₂,...aₙ be n numbers. The average or mean of a₁, a₂,...aₙ is the quantity obtained by dividing the sum of all the numbers by the total count of all. Mathematically,
Average = (a₁ + a₂ + ··· + aₙ) / n.
Let the first set of ninety numbers be a₁, a₂,...a₉₀, and the second set of sixty numbers be b₁, b₂,...b₆₀.
Given
The average of the first set of ninety numbers is 60. Mathematically,
(a₁ + a₂ + ··· + a₉₀) / 90 = 60. ...(1)
The average of the second set of sixty numbers is 90. Mathematically,
(b₁ + b₂ + ··· + b₆₀) / 60 = 90. ...(2)
Sum of all the numbers of the two sets
Calculate (a₁ + a₂ + ··· + a₉₀ + b₁ + b₂ + ··· + b₆₀).
Using equations (1) and (2) in the above sum,
(a₁ + a₂ + ··· + a₉₀ + b₁ + b₂ + ··· + b₆₀) = (60 × 90 + 90 × 60) = 2 × 90 × 60.
Average of all the numbers
Average of all the numbers = (a₁ + a₂ + ··· + a₉₀ + b₁ + b₂ + ··· + b₆₀) / (90 + 60).
Using the above value of the sum,
Average = 2 × 90 × 60 / 150 = 72.
So, the average of all the numbers is 72.
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