Author 
Topic 

TchrWill
Advanced Member
USA
79 Posts 
Posted  07/08/2013 : 15:54:04

The Krane children were born 2 years apart. When Micheal, the youngest of the Six children was 1 year old, the sum of their ages was 1+3+5+7+9+11, or 36, a perfect square. How many years old was Micheal the next time the sum of their ages was a perfect square? 


Subhotosh Khan
Advanced Member
USA
9116 Posts 
Posted  07/16/2013 : 18:35:01

quote: Originally posted by TchrWill
The Krane children were born 2 years apart. When Micheal, the youngest of the Six children was 1 year old, the sum of their ages was 1+3+5+7+9+11, or 36, a perfect square. How many years old was Micheal the next time the sum of their ages was a perfect square?
After 'n' years the sum of their ages would be
S = 36 + 6*n............The S must be divisible by 6
and that number would be 144 .... 18 years later. 


TchrWill
Advanced Member
USA
79 Posts 
Posted  07/22/2013 : 11:21:48

Right on Subhotosh Khan.
Letting M be Michael's age at any time, the sum of the six ages will next be a square when 36 + 6(M  1) = N^2. The sum of the ages increases in increments of 6 making the square we seek an even square. The sum of the ages progresses as
M's age....1....2....3…..4.…..5....6.…..7.....8...9....10 Sum........36..42…..48….54...60...66....72...78...84...90
M's age. ..11..12.…13….14...15...16....17...18...19...20 Sum........96102108114120126132138144
Looks like Michael was 19 when the sum of the ages equaled 144 = 12^2, the next square after 36.
Another way of zeroing in on the answer is to simply find what value of M will create an even square. Those above 36 proceed as 64, 100, 144, 196, 256, etc. A few punches on the old calculator shows that M = 19 satisfies N^2 = 144.
Another way of attacking this one is as follows:
1The first sum of ages is 36 at Michael's age 1. 2The subsequent sums increase by 6 each year. 3The sum sought can be defined as N^2 = 36 + 6(M  1), M being Michael's attained age. 4Written another way, N^2  36 = N^2  n^2 = 6(M  1) 5N^2  n^2 can be written as (N + n)(N  n) 6Therefore, (N + n)(N  n) = 6(M  1) 7Then, (N + n) = (M  1) and (N  n) = 6 8Adding, 2N = M + 5 or N = (M + 5)/2 9We can now write N^2 = (M  1)^2/4 = 36 + 6(M  1) 10Expanding and simplifying, M^2  14M  95 = 0 11From the quadratic formula, M = [14+/sqrt(196 + 380)]/2 12Solving, M = 19.




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