I was talking to a friend Tammy, the middle child of five. She has a younger sister Tracey, and three brothers Tommy (Oldest child), Timmy and Tony (youngest child). I asked her how old she was, but she hardly ever gives a straight answer. The following is the conversation we had.
“How old are you, Tammy?”
“I am three times as old as Timmy was when Tony was born. Actually, it’s funny, but Timmy’s, Tracey’s, Tommy’s and my ages were all the factors of Mom’s age when Tony was born. The only other year where the ages of more than two of us were the factors of Mom’s age was when Tracey was half as old as I am now.”
“How old is your Mom now?”
“There’s another funny thing. When Mom turns 50, Tony will be the same age Mom was when she gave birth to Tommy.”
Assuming Tammy was talking in terms of whole years, and with my mathematical and reasoning abilities I was able to deduce quickly not just Tammy’s age, but all of their ages. So how old are Tommy, Timmy, Tammy, Tracey, Tony and Mom now?
Answer
Tommy is 23, Timmy is 19, Tammy is 18, Tracey is 16, Tony is 13 and Mom is 43.
Solution:
x = Mom’s age when she gave birth to Tommy = Tony’s age when Mom turns 50;
y = Tommy’s age when Tony was born
z = Mom’s age when Tony was born
From Tammy’s last statement, 2x + y = 50 and it can be deduced that x = z – y, where y is a factor of z. By combining the two equations you get 2z = 50 + y. The only possible solutions to that equation so that y may also we a factor of z are (z = 26, y = 2) and (z = 30, y = 10). Only 30 has four or more factors, so Mom was 30 and Tommy was 10 when Tony was born.
This means that of the remaining factors of 30, Timmy was 6, 5 or 3, Tammy was 5, 3 or 2 and Tracey was 3, 2 or 1. This makes Tammy 18, 15 or 9 now. The only other time where the ages of more than two of the children were the factors of Mom’s age was when Tracey was half what Tammy is now, so Tammy must be 18 now.
In the second multiple-factors-of-Mom’s-age year, Tracey was therefore 9. When Tony was born, Tracey was either 3, 2 or 1, giving possible time differences of 6, 7 or 8 years, which correspond to Tony’s age in the second multiple-factors-of-Mum’s-age year. It also means Mom was 36, 37 or 38 that year. 37 is prime and 38 has factors of 1, 2 and 19, and as Tony was at least 6, it means that Mom must have been 36 that year.
Going back to when Tony was born we have that Mom was 30, Tommy was 10, Timmy was 6, Tracey was 3, and as the middle child Tammy must have been 5. If Tammy is 18 now then Tony is 13, Tracey is 16, Timmy is 19, Tommy is 23 and Mom is 43.