The two numbers are 4 and 6.
Of course you could guess and check which two numbers make 10 (e.g. 1&9, 2&8,...5&5) but the statement isn't clear if the number is positive, whole, fraction, rational, etc. which could take forever so let's derive using math and not guess.
So the two statements give two equations, which should be enough to figure out the number. First equation is X+Y=10. Second equation is (1/X)+(1/Y)=(5/12). Taking the first equation and rearranging to one variable, we get X=10-Y. Substituting the rearranged first equation into the second equation, we get (1/(10-Y))+(1/Y)=(5/12).
The revised second equation does not have a common denominator so we'll need to multiple by a 'theoretical 1' to get a common denominator. The first term will need to be multiplied by (Y/Y) and the second term will be multiplied by ((10-Y)/(10/Y)). The right hand side (5/12) remains the same since we technically multiplied the left side individual terms by a theoretical 1. The left side is now (Y+(10-Y))/(-Y2+10Y). Simplifying the numerator, we get 10. The equation now reads 10/(-Y2+10Y)=(5/12). Cross multiplying, we get 120 = -5Y2 + 50Y. This is a quadratic equation (i.e. single variable squared). Rearrange to get all terms on one side and zero on the other side. Thus 5Y2 - 50Y - 120 = 0. Use the quadratic equation (a=5, b=-50, c=120) and solve for Y. Thus, Y is equal to 4 and 6.
Going back to the original equation, plug in Y to the easier equation. Let's assume Y=4. Then we quickly see X=6 (based on the first simple equation). If we assumed Y=4, then X=4. Essentially no difference.
We need to check the second equation to truly verify our answer. (1/4)+(1/6) need to have common denominators so 12 is the best LCM (Lowest Common Denominator). Multiply each term by their respective 'theoretical 1' so (3/3)*(1/4) + (2/2)*(1/6) is equal to (3+2)/12, which is 5/12.