An age problem

Let the present age of Jill be $x$ and Jack is $n$ years older. So Jack's present age is $x+n$ .

When Jack was Jill's present age that is $x$ years old, Jill's was $x-n$ years old. Therefore $x=2(x-n)$ $\implies x = 2x-2n$ $\implies x = 2n$ .

When Jill gets to be Jack's age that is $x+n$ , then Jack will be $x+n+n$ years old. Therefore, the sum is $2x+3n=63$ . Since $x=2n$ , we have $7n=63$ $\implies n = 9$ .

So the sum of their present ages is $2x+n=5n=\boxed{45}$ .

Letting $a$ be Jill's present age, the given conditions would be as shown in the tabulation.

Comparing the past and present ages of Jill imply that $\frac{a}{2}$ years have passed by, and hence that the present age of Jill would be $a+\frac{a}{2}$ = $\frac{3a}{2}$ , filling up the blank space in the table.

Now, when Jill gets to be Jack's age implies that the future age of Jill would be the same to the present age of Jack.

Another $\frac{a}{2}$ years have passed by from the present up to the future, implying Jack's age would be $\frac{3a}{2}$ + $\frac{a}{2}$ $= 2a$

But the sum of their ages when Jill gets to be Jack's age is 63; that is,

$2a + \frac{3a}{2}$ = 63

or $a = 18$

Finally, the sum of their present ages would be $18 + 18* \frac{3}{2}$ , or 45 .

When Jack was as old as Jill's present age, we let this age be $x$ . Let's say this happened $n$ years ago. Considering that Jill is now $x$ years old, she would be $(x-n)$ years old back then. With Jack's age at that time as $x$ years old, we see that:

$x=2(x-n)$

$x=2x-2n$

$x=2n$

In addition, if Jack was $x$ years old $n$ years ago, he would now be $(x+n)$ years old. Now, Jill is $x$ years old so Jack will always be $n$ years older than Jill. Now when Jill becomes $(x+n)$ years old, Jack would be $x+n+n=x+(1+1)n=x+2n$ years old.

Now their combined age would be $(x+2n)+(x+n)=(1+1)x+(2+1)n=2x+3n$ which is given to be $63$ . Considering that $x=2n$ as determined, we see that:

$2(2n)+3n=63$

$(2\times 2)n+3n=63$

$4n+3n=63$

$(4+3)n=63$

$7n=63$

$n=9$

If Jack is now $(x+n)$ years old and Jill, $x$ years old, they total $2x+n=2(2n)+n=(2\times2)n+n=4n+n=(4+1)n=5n=5\times 9 =\boxed{45}$ years of age.