Angelo Canedo Figures Out Manny's Age

It took a former SLC Engineering instructor to solve the second problem posed on my previous post titled An Amusing Challenge. Engr. Angelo Canedo belongs to the the cream of the crop of SLC CE batch 1996. He now works for a multinational structural steel producer based in the UAE.

How old is Manny?

I have two friends named Manny and Danny who are very conscious of their ages. They don’t only keep tract of their years, they also are aware of their age combinations and ratios along the way. You see, the combined ages of Manny and Danny is 80 years, and Manny is twice as old as Danny was when Manny was half as old as Danny will be when Danny is three times as old as Manny was when Manny was three times as old as Danny. So how old is Manny?

Angelo's Solution

Let
M be Manny’s age and D be Danny’s age
x = be the year interval from present year P to year A (see line diagram)
y = be the year interval from year A to year B (see line diagram)
z = be the year interval from year B to year C (see line diagram)
P = be the present year (see line diagram)



Equations:

combined ages of Manny and Danny is 80

M + D = 80 …………………………………. Equation 1

and Manny is twice as old as Danny was …….

M = is Manny’s present age at P
D – x = was Danny’s age x yrs ago from P to A

M = 2 ( D – x ) ……………………………… Equation 2

….. when Manny was half as old as Danny will be ……

M – x = was Manny’s age x yrs ago from P to A
D – x + y = will be Danny’s age y yrs after from A to B

M – x = ½ ( D – x + y ) ……………………… Equation 3

….. when Danny is three times as old as Manny was ……

D – x + y = is Danny’s age at B
M – x + y – z = was Manny’s age z yrs ago from B to C

D – x + y = 3 ( M – x + y – z ) ……………… Equation 4

….. when Manny was three times as old as Danny.

M – x + y – z = was Manny’s age z yrs ago from B to C
D – x + y – z = was Danny’s age z yrs ago from B to C

M – x + y – z = 3 ( D – x + y – z ) …………… Equation 5

Combining the 5 equations and solving them simultaneuosly, Manny’s age is 50.

Hehehe... Ganon lang pala yon kasimple. The critical point here is to realize that the ages of Manny and Danny are being compared across four points in time. Once this fact is grasped, the puzzle unravels surprisingly easy.