# “Solution to” vs. “solution of ”

What is the difference between saying solution to the problem and saying solution of the problem?

Are they both equivalent, or is there some difference?

• Related: solution for or to a problem? May 7, 2011 at 0:51
• similar, but "for" is not "of" :-) Maybe we should combine the two questions... May 7, 2011 at 1:03
• The answer to the other question is the same for this question. May 7, 2011 at 3:00
• Is `2` the solution to the equation `x-2=0`, or is it a solution of the equation?
– SF.
Oct 21, 2012 at 21:52

You almost always hear "solution to the problem" and sometimes "solution for the problem" — but almost never do you hear "of" in that context.

solution |səˈloō sh ən| noun 1 a means of solving a problem or dealing with a difficult situation : there are no easy solutions to financial and marital problems. • the correct answer to a puzzle : the solution to this month's crossword.

Note that both of NOAD's examples use to.

One would use "solution of" if one is referring to a chemical solution:

2 a liquid mixture in which the minor component (the solute) is uniformly distributed within the major component (the solvent).

• Hmm: google.com/… May 7, 2011 at 4:55
• Following on from @mplungjan... there's a difference in meaning between The solution to that problem was satisfying, and The solution of that problem was satisfying - i.e. whether it's the result or the process that was satisfying. May 7, 2011 at 8:17
• How about an equation? "Find the solution of x+2=4" vs. "Find the solution to x+2=4" ? Nov 21, 2013 at 2:03
• I would still prefer to there. Nov 21, 2013 at 3:04

It depends on the context. The main distinction is that solution has more than one meaning:

• An answer to the problem (fully worked out), but also
• The process of finding such an answer (i.e. a synonym for solving)
• Something dissolved in something else (say, sugar dissolved in water)

With the first meaning, use to:

I have found a solution to your problem.

With the second, use of:

The solution of those problems took me a long time.

The solution of difficult problems is very satisfying.

Note that choice of to versus of can change the meaning:

The solution to that problem will take you an hour. [Finding the solution took a year, but carrying it out takes an hour.]

The solution of that problem will take you an hour. [It will take an hour to solve it.]

When talking about something dissolved in something else (usually in chemistry), use of:

This drink is basically a solution of sugar in water.

It depends heavily on the context.

In colloquial usage you almost always (from personal experience) hear solution to the problem, maybe sometimes solution for the problem (As in - "I have a solution for/to your problem")

In mathematics however, "Solution of" can be used as well (e.g. "A solution of the differential equation" however "A solution to the system of differential equations"). Perhaps the tense of the object, and the fact that it can possess solutions (rather than be solved by solutions as above) allows one to use the "of" form.

For me, there does exist a slight difference between saying 'solution to' and 'solution of'.

When using the 'of' form, we are treating the solution as an attribute of that problem, i.e. we are more confident about the correctness of the solution.

When using the 'to' form, we are relatively less confident about that, sometimes we are just being in the middle of the process of proving the correctness of the solution.

The only usage of 'solution of' that seems right to be is in the chemical sense. E.g. "A weak solution of calcium chloride in water."