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| The Trent And Newley Families (Posted on 2006-08-31) |
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Last month, the Trent family moved next to the Newley's. One fine afternoon, Mrs. Trent called on Mrs. Newley to get acquainted with the Newley family. In the course of the conversation, when Mrs. Trent queried Mrs. Newley about the age of her daughter Jackie, Mrs. Newley responded cryptically, “When she was born, I was precisely five times the age of her brother Tony at that time. Six years from now, Jackie will be one third of the age which I will attain at that time.”
At this point, Mrs. Trent requested some additional information when Mrs. Newley replied, ”The sum of squares of the digits of my husband’s present age in years, is two more than my current age. Presently, my age is nine years more than that of Tony’s father (my husband) when Tony was born”.
Determine the current ages of Mr. Newley, Mrs. Newley and their two children from the above statements.
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Submitted by K Sengupta
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Rating: 4.5000 (2 votes)
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Solution:
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The respective ages of Mr. Newley, Mrs. Newley, Jackie and Tony are 45 years, 39 years, 9 years and 15 years.
EXPLANATION:
Let the respective ages (in years) of Mr. Newley, Mrs. Newley, Jackie and Tony be M, F , G and B respectively. We denote sum of squares digits of a given number by s.s.d.
Then, by conditions of the problem:
(1) M – G = 5 (B – G);
(2) M – 9 = F – B;
(3) M + 6 = 3 (G+6)
(4) s.s.d. (F) = M +2
M – 3G = 12 ( from (3) ) and M + 4G = 5B (from (1))
This gives, 5B – 7G = 12-----------(5)
Since (B,G) = (8,4) is an integral solution for (5), we can take B = 7k + 8, where k must be a non negative integer, giving G = 5k + 4; so that M = 3G +12 = 15k + 24.
Accordingly, from (2); F = M+B– 9; giving, F = 22k + 23, so that:
(M, F) = (15k +24, 22k +23); where k is a non negative integer.
Consequently, (M, F) = (24,23); (39, 45); (54, 67); (69, 89) and (84, 111) are the only available choices keeping in mind the normal life span of a human being.
Of the above choices, only (M, F) = (39, 45) satisfies condition (4), since:
4^2 + 5^2 = 41 = 39 +2.;
so that k = (39-24)/15 =1
It can easily be verified that Condition (4) is not satisfied for any other choices of (M, F).
Therefore, G = 5*1 + 4 = 9 and B = 7*1 + 8 = 15.
Consequently, the respective ages of Mr. Newley, Mrs. Newley, Jackie and Tony are 45 years, 39 years, 9 years and 15 years.
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For an alternate method, refer to Charlie's solution provided in this location.
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