### Brain Teasers

# Tammy's Age

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 factors of Mum's age when Tony was born. The only other year where the ages of more than two of us were factors of Mum's age was when Tracey was half as old as I am now."

"How old is your Mum now?"

"There's another funny thing. When Mum turns 50, Tony will be the same age Mum 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 quickly deduce not just Tammy's age, but all of their ages. So how old are Tommy, Timmy, Tammy, Tracey, Tony and Mum now?

"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 factors of Mum's age when Tony was born. The only other year where the ages of more than two of us were factors of Mum's age was when Tracey was half as old as I am now."

"How old is your Mum now?"

"There's another funny thing. When Mum turns 50, Tony will be the same age Mum 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 quickly deduce not just Tammy's age, but all of their ages. So how old are Tommy, Timmy, Tammy, Tracey, Tony and Mum now?

### Answer

Tommy is 23, Timmy is 19, Tammy is 18, Tracey is 16, Tony is 13 and Mum is 43.Solution:

x = Mum's age when she gave birth to Tommy = Tony's age when Mum turns 50;

y = Tommy's age when Tony was born

z = Mum'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 for that equation so that y is also a factor of z are (z = 26, y = 2) and (z = 30, y = 10). Only 30 has four or more factors, so Mum 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 factors of Mum's age was when Tracey was half what Tammy is now, so Tammy must be 18 now.

In the second multiple-factors-of-Mum'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 Mum 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 Mum must have been 36 that year.

Going back to when Tony was born we have that Mum 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 Mum is 43.

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