This puzzle is the eleventh puzzle from the Matt Parker's Maths Puzzles series.
David and Anton's ages combined equals 65. David is currently three times as old as Anton was when David was half as old as Anton will be when Anton is three times as old as David was when David was three times as old as Anton.
How old is David?
Let d(t) and a(t) be David's and Anton's ages at time t respectively where
t = 0 is the current time and the past is indicated by t < 0 and the future
is indicated by t > 0. The first sentence gives us the relation d(0) + a(0) = 65. Since we can assume that they both age at the same rate, we have:
a(t) = t + a0d(t) = t + d0
Where a0 and d0 are the current
ages of David and Anton respectively. We can now attempt to unpack the second
sentence.
The last part of the second sentence says
when David was three times as old as Anton.
So for some t1 < 0, d(t1) = 3 •
a(t1). This time is used to set David's age against Anton's.
when Anton is three times as old as David was [at time
t1].
As beforewe have some t2 > 0 such that
a(t2) = 3 • d(t1). We then use this time for
next part.
when David was half as old as Anton will be [at time
t2].
Giving us some t3 < 0 such that d(t3) =
1∕2 • a(t2). Which we use in the final part.
David is currently three times as old as Anton was [at time
t3].
And we have d(0) = 3 • a(t3).
The equations and inequalities we have in terms of different t give us the
following equations and inequalities in terms of a0 and
d0:
| Original |
In terms of a0 and d0
|
|---|---|
d(t1) = 3 • a(t1)
|
t1 = (d0 - 3 • a0)∕2
|
a(t2) = 3 • d(t1)
|
t2 = (9 • d0 - 11 • a0)∕2
|
d(t3) = 1∕2 • a(t2)
|
t3 = (5 • d0 - 9 • a0)∕4
|
d(0) = 3 • a(t3)
|
d0 = 15∕11 • a0
|
Using the first fact that d0 + a0 = 65 we
have
d0= 37.5a0= 27.5
We can also calculate the different times:
t1= -22.5; David was 15 and Anton was 5.t2= 17.5; David will be 55 and Anton will be 45.t3= -15; David was 22.5 and anton was 12.5.
Something that irritates me is the fractional age, since if David has spent 37.5 years alive, then he will only be 37 years old. Likewise Anton would only be 27, and their combined age would only be 64. This however does not play nicely with the conditions in the question, so perhaps it is unwise to go too far beyond the scope.