Another perspective
Hi. You seem to have had your question suitably answered but since I had already written this I will post it. It might give you a different perspective and that is always valuable.
To see the relationships which JeffM details consider two buildings standing next to each other, a ten story building and a 7 story building.
A key thing is to immediately see is a third important quantity, the absolute difference in height, which is three stories.
What questions can then be asked? Compare the height of the two buildings? Is that a fair question, not really, it is a ambiguous, compare the absolute height, or the relative difference, and with respect to which building?
1) Compare the absolute height of the 10 story building to the 7 story building.
But wait you might say, are you asking me to make the comparison with respect to the the 10 story building or with respect to the seven story building?
It is easy to see and say that the 7 story building is 70% as tall as the 10 story building. The calculation is (7/10)(100%) with the standard of comparison, the 10, in the denominator.
Less obvious is that the 10 story building is 142.8...% as tall as the 7 story building. The calculation is (10/7)(100%) with the standard of comparison, the 7, in the denominator.
2) Compare the relative difference of the height between the two buildings.
There again one might say ... but wait, with respect to the 10 or the 7 story building? The difference, as we previously and immediately noticed is 3 stories. The calculations parallel those above.
(3/10)(100%) = 30 % The 7 story building is 30% of the 10 story building shorter, the standard of comparison, the 10, is in the denominator.
(3/7)(100%) = 42.8...% The 10 story building is 42.8 % of the 7 story building taller, the standard of comparison, the 7, is in the denominator.
To emphasize the importance of keeping track of what a percentage figure is referring to consider this, if you shrink the 10 story building by 30% so that it is equal in height to the 7 story building, in order to grow it back to 10 stories you will need to increase its height by 42.8% of its then existing height. (A sad fact when you looking up from a down turn in the economy.)
With regard to your actual problem it is not clear to me whether you are asking for the absolute or relative percent increase from 38K to 50K. The way the problem is stated I would have thought the former (50/38)(100%) =131.6% but the way you started the problem seems to indicate the latter.
So, what were you thinking, where did you go wrong. You started off with the example of 100, and, essentially said divide it into two parts, 70 and 30, Then, after (mental) calculation determined that the 70 part is 70% and so the remainder of 30 must represent 30%. Fine.
So then it would be ok to divide 50K into two parts, a 38K part and rest. Fine. Calculation this time shows that 38K is 76% of 50K. Fine. Then, unfortunately you thought that this meant that because the remainder is 24% of 50 you should be able to multiply 38K by 1.24 and get back to 50K thus proving that 50k is a 24% increase over 38K. Didn’t work. Why not? For the same reason it didn’t work in your first example. 70(1.3) = 91 not 100. !!!
Hate when that happens, you assume a bad premise and spends hours trying to find an error in your logic. The error in premise is illustrated in the example of the shrinking building above, the percent loss is based on quantity A, the percent recovery must be based on quantity B
I am not a math specialist, in spending time with your problem I learned first, keep the facts straight, the quantity A, the quantity B, and the quantity C (the positive difference between A and B, and which I must remember to IMMEDIATELY TABULATE to keep myself clear about what I doing). Second, keep straight which quantity you are using as the standard of reference, A, or B, (or C if you want to ponder that).
