The ratio 1:3 becomes 1:4 after the $68 purchase by both people.

Because the difference in their money remains unchanged even after purchasing the racket, we change the ratio into a form such that the difference between the two numbers in each ratio is the same.

1:3 has a difference of 3 - 1 = 2.
1:4 has a difference of 4 - 1 = 3.

We make the difference equal to 6 to reflect the case that the difference in money is the same at 6 units.

1:3 is to be changed into 3:9.
1:4 is to be changed into 2:8.

So the expenditure of Badminton racket of $68 is 1 unit, since each of them drops in unit by 1.

1 unit = $68
3 units = $204

So Jack has $204 at first.

Check.

9 units = $612 so the ratio 204:612 = 1:3.

204 - 68 = 136
612 - 68 = 544

so the ratio 136:544 = 1:4.

Thank you very much for your explanation.
May I know if both spent different amount, can I use this method?
Example if Jack spent 60 & john spent 76.

The nice thing about this method is that it works very well for same differences. It is not as straightforward if the values are different.

In other words, this method does not work for different values.

Well noted with thanks

Eric Nicholas K you are smart and reasonable. Thank for your explanation