Given that Set H contains five positive integers such that the mean, median, mode, and range are all equal. The sum of the data is 25. And we need to find the largest possible number in set H

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Theory

‣‣‣ Mean = (Sum Of All The Numbers) / (Total Number Of Numbers)
‣‣‣ Mode is the number which has occurred the maximum number of times in the set.
‣‣‣ Median is the middle value of the set
‣‣‣ Range of a set is the difference between the highest and lowest value of the set.

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As there are 5 numbers so Median = Middle term = Third Term
Mean = \(\frac{Sum}{5}\) = \(\frac{25}{5}\) = 5
Mean = Median = Mode = Range = Third Term = 5

So, the set is _ , _ , 5 , _ , _

Now, the mode is 5 so 5 has to occur the maximum number of times.
Let's say 5 occurs 2 times so we have two possibilities

_ , 5 , 5 , _ , _
_ , _ , 5 , 5 , _

In, _ , 5 , 5 , _ , _ case we will have two numbers bigger than 5. Lets say 6 and 7 and range is 5 so first number will become 7-5 = 2
So the set becomes 2, 5, 5, 6, 7 and it satisfies all conditions

_ , _ , 5 , 5 , _ case lets take the two smaller numbers as 3 and 4. So the largest number will become 3 + 5 = 8
So the set becomes 3, 4, 5, 5, 8 and it satisfies all conditions

Clearly Quantity A(10) > Quantity B(7 or 8)

So, Answer will be A.
Hope it helps!

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