The product of two numbers is 2160 and their HCF is 12. If the sum of the squares of the two numbers is 4896, then what is the mean of the two numbers ?
48
Concept:
Average = Sum of observations / Total number of observations
Highest Common Factor (HCF) = The smallest number which is completely divisible by two or more numbers.
Calculation:
Consider the numbers are 12a and 12b.
According to the question,
12a × 12b = 2160
144ab = 2160
ab = 15 ------(1)
Again, the sum of the squares of the two numbers is 4896.
(12a)2 + (12b)2 = 4896
144(a2 + b2) = 4896
a2 + b2 = 34 ------(2)
We know that,
(a + b)2 = a2 + b2 + 2ab
(a + b)2 = 34 + 2 × 15 = 64
a + b = 8 -------(3)
Therefore, the average of the two numbers
(12a + 12b)/2 = 6(a + b)
From equation (3),
Average = 6 × 8 = 48
∴ Required average = 48
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