Mark, Mike, Joe, Jim and Dan are brothers. All of their ages are different. The sum of their ages is 95. Dan is 15 years old and is the youngest. Mike is the second oldest. What is the maximum possible age for Mike?
The sum of the 5 brothers' ages is 95 years.
We know that Dan is 15 years old, so we can subtract 15 from 95.
95 - 15 = 80
We know that the sum of the ages of the remaining 4 brothers is 80 years.
Because we are working with the sum of the brothers' ages, in order to maximize someone's age (we are maximizing Mike's age in this case) we have to minimize the rest of the brothers' ages.
Since Mike has two younger brothers after Dan, we will minimize their ages.
Joe and Jim cannot be the same ages as each other or as Dan, which means that the smallest possible ages they can be are 16 and 17 years old, so we will assign those ages to Joe and Jim.
Jim = 16; Joe = 17
We can now subtract the total ages of Joe and Jim so that the remaining number is the sum of the age of Mark and Mike.
80 - (16 + 17) = 47
Now that Mike is the younger of the two siblings remaining and we still want him to be as old as possible, we will make Mike and Mark as close in age as possible. Specifically, they will be one year apart.
As Mark gets older, Mike will have to get younger in order to maintain that the sum of their ages is still 47. This means that in order for Mike's age to be maximized, he must be exactly one year younger than Mark.
We can define Mike's age as x and Mark's age as (x + 1)
x + (x + 1) = 47
2x + 1 = 47
2x = 46
x = 23
Answer: Mike is 23 years old
