Johnny is three times as old as his son was when....

[Jess]

Johnny is three times as old as his son was when Johnny was twice as old as his son will be two years from now. What is Johnny's age now?

 

[Denis]

Johnny(son) ; take your pick:
21(8), 30(12), 39(16), ..... , 93(40), 102(44), ........ , 993(440), 1002(444):biblical days!, .....

 

[sgtpepper]

Frankly, I don't think you can solve this without guessing and checking. Because, you are basically solving for 4 variables
if you assigned the following:
x = Johnny's son's age (JS)
y = Johnny's previous age some unknown amount of time ago, when it equaled his son's present age plus 2 and times 2 (JP)
z = Johnny's age now (J)
w = Johnny's son's previous age with respect to Johnny's old age, y (JSP)

it would look like this:

2(x + 2) = y
z = 3w

or:

2 times (JS plus two) = JP
J = 3 times JSP

so, I'm pretty sure you can't solve this without guessing and checking

 

[stapel]

Johnny is three times as old as his son was when Johnny was twice as old as his son will be two years from now. What is Johnny's age now?

Work in stages.

. . .ages now:
. . . . .Johnny: j
. . . . .son: s

. . .ages two years from now:
. . . . .Johnny: j + 2
. . . . .son: s + 2

We know that Johnny "is" (j) three times as old as his son "was" back when. Back when, Johnny's age was twice his son's age two years from now. That is:

. . . . .j - ?? = 2(s + 2)

Whatever that difference, ??, was, Johnny currently is three times his son's age then:

. . .ages back when:
. . . . .Johnny: j - ??
. . . . .son: s - ??

And:

. . . . .j = 3(s - ??)

So we have:

. . . . .j - ?? = 2(s + 2)
. . . . .j = 3(s - ??)

Then:

. . . . .3(s - ??) - ?? = 2(s + 2)
. . . . .3s - 3(??) - 1(??) = 2s + 4
. . . . .3s - 4(??) = 2s + 4
. . . . .s - 4(??) = 4
. . . . .s - 4 = 4(??)

Since the unknown difference is some whole number, and since we're assuming that the son had actually been born as of two years ago (so his age was something more than "zero years"), then "s" must be 5 or more (so 4(??) is a positive value, so ?? is a positive whole number).

So pick values for the difference, ??, between "now" and "back when", and see what works.

Eliz.

 

[soroban]

Hello, Jess!

Is there more to the problem?
. . As stated, there is are many solutions.

Johnny is three times as old as his son was when Johnny was twice as old
as his son will be two years from now. .What is Johnny's age now?

I set up my famous chart . . .

Let \(\displaystyle J\) = John's age now.
Let \(\displaystyle S\) = his son's age now.

        |  Now  |           |             |
- - - - + - - - + - - - - - + - - - - - - +
  John  |   J   |           |             |
- - - - + - - - + - - - - - + - - - - - - +
   Son  |   S   |           |             |
- - - - + - - - + - - - - - + - - - - - - +

\(\displaystyle y\) years ago, they were both \(\displaystyle y\) years younger.
. . John was \(\displaystyle J\,-\,y\) years old.
. . His son was \(\displaystyle S\,-\,y\) years old.

        |  Now  | y yrs ago |             |
- - - - + - - - + - - - - - + - - - - - - +
  John  |   J   |   J - y   |             |
- - - - + - - - + - - - - - + - - - - - - +
   Son  |   S   |   S - y   |             |
- - - - + - - - + - - - - - + - - - - - - +

Two years from now, they will both be two years older.
. . John will be \(\displaystyle J\,+\,2\) years old.
. . His son will be \(\displaystyle S\,+\,2\) years old.

        |  Now  | y yrs ago | 2 yrs hence |
- - - - + - - - + - - - - - + - - - - - - +
  John  |   J   |   J - y   |    J + 2    |
- - - - + - - - + - - - - - + - - - - - - +
   Son  |   S   |   S - y   |    S + 2    |
- - - - + - - - + - - - - - + - - - - - - +

We are told that John is (now) 3 times as old as his son was \(\displaystyle y\) years ago.
. . We have: \(\displaystyle \L\:J\:=\:3(S\,-\,y)\)

We are told that, at that time, John was 2 times as old as his will be 2 years hence.
. . We have: \(\displaystyle \L\:J\,-\,y\:=\:2(S\,+\,2)\)


We have a system of two equations in three variables.
. . There is no unique solution.