How can I find the a certain birthdate of two people when one information given depends on the other?

Question

The problem is as follows:

Mike was born on $\textrm{October 1st, 2012,}$ and Jack on $\textrm{December 1st, 2013}$. Find the date when the triple the age of Jack is the double of Mike's age.

The alternatives given in my book are as follows:

$\begin{array}{ll} 1.&\textrm{April 1st, 2016}\\ 2.&\textrm{March 21st, 2015}\\ 3.&\textrm{May 8th, 2015}\\ 4.&\textrm{May 1st, 2015}\\ \end{array}$

I tried all sorts of tricks in the book to get this one but I can't find a way to find the given date. What sort of formula or procedure should be used to calculate this date? Can someone help me?

Answer 1

We have $M=14+J$, where $M$ is Mike's age in months and $J$ is Jack's age in months,

and $2\times M=3\times J$. Substitute $14+J$ for $M$ in that last equation and solve for $J$.

Then you know how old Jack is when $2\times M=3\times J$,

and from that, with the date of Jack's birth, you can figure the date when $2\times M=3\times J$.

Answer 2

So first, do some math (or ask Google or Wolfram or something) to find how many days after Mike's birthday Jack's is. The answer is $426$ days. Therefore, if $J$ is Jack's age, then $M$ (Mike's age) satisfies $M=J+426$.

We seek the date when "triple Jack's age" is "double Mike's age," i.e. $3J = 2M$.

Substitute $M=J+426$, and then solve the equation you get for $J$. $J$ can be found to be $852$ this way.

Therefore, when Jack is $852$ days old, the condition is satisfied. (That is, $852(3) = 2(852+426) = 2556$.)

Thus, you need to figure out what date is $852$ days after Jack's birth. You can, again, do this by hand, or just ask some web service to determine it is April 1, 2016.

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Granted, this just seems like the obvious solution. Maybe there is some trick involved, but I don't see it, and sometimes the solution is just the obvious one.

Answer 3

Just do it.

Mike was born Oct 1st, 2012. So $365$ days later is Oct 1st, 2013. And $31$ days after that is Nov. 1st, 2013 and Mike is $365+31 = 396$ days old. And $30$ days after that is Dec. 1st, 2013. Mike is $396+30=426$ days old and Jack is born.

So Mike is $426$ days older than mike.

So we want to find when $3J = 2M$ where $J = $ Jack's age in days and $M$ is Mike's age in days. ANd we know $M = J +426$.

So $3J = 2(J + 426)$

$3J = 2J + 852$.

So $J = 852$. So we need to find the date when $J$ is $852$ days old.

On Dec. 1, 2014, Jack is $365$ days old.

On Dec. 1, 2015, Jack is $730$ days day old.

Jan 1. 2016 is $31$ days later and Jack is $761$ old.

Feb 1. 2016 is $31$ days later and Jack is $792$ days old.

March 1. 2016 is $29$ days later (2016 is a leap year) and Jack is $821$

April 1. 2016 is $31$ days later and Jack is $852$ days old.

That's it.

If you are in a hurry....

One year and two months is $\approx 1\frac 2{12}$ years.

So we have $M = J + 1\frac 2{12}$ and need $3J = 2(M +1\frac 2{12})= 2M + 2\frac 4{12}$ so so $J = 2$ years and $4$ months approximately.

And $2$ years and $4$ months after Dec 1. 2013 is April 1. 2016. This may be off by a few days as not all months have the same numbers of days.

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Perhaps the most adjustable and pragmatic idea is distinguish:

Let $D$ be the difference in ages so $M = J + D$.

We need to find the date when $3J = 2M = 2(J+D)$ so $3J = 2J + 2D$ so $J = 2D$.

That is when $J$ is twice as old as the difference between them.

As $D$ is about a little over a year we need a time when Jack is about two years old and somewhat more. $A$ is the only one even remotely close.

I suppose the trick is to realize that as Jack was born in Dec, there very end of 2013, so do not just subtract 2013 from 2015 or 2016 and figure about 2 or 3 years. They are instead about a year and a bit and two years and a bit. As we need two years and four months. April 1, 2016 is the closest.

And if you want to figure out $D$ in detail its $1$ year and $2$ months more or less, $426$ days exactly, $61$ weeks roughly, $10,000$ plus hours, or whatever.