Problema Solution
Ted emptied an envelope on to the table,
" Here's 99 stamps your Uncle Jim sent for you two kids, he said.
He wants them shared in a crazy way. First you,Tony, take 4 for each year of your age.
Henry's younger,so he takes 3 for each year. Then the two of you split the rest equally."
It didn't take long to do,and then little Henry exclaimed: " Look,i got 4in all for every year of my age."
What were the ages?
Answer provided by our tutors
let
t = the age of Tony, h is integer
h = the age of Henry, r is integer
t > h
4t + 3h + r = 99, where r is the reminder, r is integer
From Henry's words " Look,i got 4 in all for every year of my age." we have
3h + r/2 = 4h
h = r/2
r = 2h
plug r = 2h into 4t + 3h + r = 99
4t + 3h + 2h = 99
4t + 5h = 99
t = (1/4)(99 - 5h)
we will assume that Henry is at least 4 years of age (since he knows how to add)
for h = 4
(1/4)(99 - 5*4) is not an integer
for h = 5
(1/4)(99 - 5*5) is not an integer
for h = 6
(1/4)(99 - 5*6) is not an integer
for h = 7
(1/4)(99 - 5*7) = 16
Henry is 7 years old and Tony is 16 years old.
for h = 8
(1/4)(99 - 5*8) is not an integer
for h = 9
(1/4)(99 - 5*9) is not an integer
for h = 10
(1/4)(99 - 5*10) is not an integer
for h = 11
(1/4)(99 - 5*10) = 11 but the age of Tony must be bigger then the age thus this is not a solution.
if we increase the value for h, the value for t is decreasing but the condition says that t>h and this is where we stop.
The solution is Tony is 16 years old and Henry is 7 years old.
Indeed
4*16 + 3*7 + 14 = 99
3*7 + (14/2) = 28 = 4*7.