A person is said to be n years old ( where n is a non−negative integer) if the person has lived at least n years and has not lived n+1 years. At some point Tom is 4 years old and John is three times as old as Mary. At another time, Mary is twice as old as Tom and John is five times as old as Tom. At a third time, John is twice as old as Mary and Tom is t years old. What is the largest possible value of t?


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Question Number 517 by Yugi last updated on 25/Jan/15

$A p e r s o n i s s a i d t o b e n y e a r s o l d (w h e r e n i s a n o n - n e g a t i v e i n t e g e r) i f$ $t h e p e r s o n h a s l i v e d a t l e a s t n y e a r s a n d h a s n o t l i v e d n + 1 y e a r s . A t s o m e p o i n t$ $T o m i s 4 y e a r s o l d a n d J o h n i s t h r e e t i m e s a s o l d a s M a r y . A t a n o t h e r t i m e,$ $M a r y i s t w i c e a s o l d a s T o m a n d J o h n i s f i v e t i m e s a s o l d a s T o m . A t a t h i r d$ $t i m e, J o h n i s t w i c e a s o l d a s M a r y a n d T o m i s t y e a r s o l d . W h a t i s t h e l a r g e s t$ $p o s s i b l e v a l u e o f t ?$
Commented by prakash jain last updated on 23/Jan/15

$T, J, M a r e a g e s o f T o m, J o h n a n d M a r y$ $p_{1}$ $T_{1} = 4$ $J_{1} = 3 M_{1}$ $p_{2} L e t u s s a y k y e a r s f r o m p_{1}$ $T_{2} = 4 + k$ $M_{2} = 2 T_{2} = 8 + 2 k$ $J_{2} = 5 T_{2} = 20 + 5 k$ $M_{1} = 8 + 2 k - k = 8 + k$ $J_{1} = 20 + 5 k - k = 20 + 4 k$ $M_{1} = 8 + 2 k - k = 8 + k$ $J_{1} = 3 M_{1} \Rightarrow 20 + 4 k = 24 + 3 k \Rightarrow k = 4$ $M_{1} = 12, J_{1} = 36, M_{2} = 16, J_{2} = 40, T_{2} = 8$ $p_{3} L e t u s s a y l y e a r s f r o m p_{1}$ $T_{3} = t = 4 + l$ $J_{3} = 2 M_{3}$ $J_{3} = J_{1} + l = 36 + l$ $M_{3} = M_{1} + l = 12 + l$ $36 + l = 2 (12 + l)$ $36 + l = 24 + 2 l \Rightarrow l = 12$ $S o t = 4 + 12 = 16$
	Answered by prakash jain last updated on 23/Jan/15

	$S e e c o m m e n t o n l y o n e s o l u t i o n t = 16$
	Commented by prakash jain last updated on 23/Jan/15

	$T_{3} = t = 16$ $M_{3} = 24$ $J_{3} = 48$
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