I don't know how to make a formula for this word problem: 
Two numbers are in the ratio of 7 to 10. If 24 is added
to each number, the resulting numbers are in the ratio 
of 1 to 2. Find the smaller of the two original numbers.

Two numbers are in the ratio of 7 to 10

 = 

which becomes upon cross-multiplying:

 = 

If 24 is added to each number, the resulting numbers
 are in the ratio of 1 to 2

 = 

which becomes upon cross-multiplying:

 = 

 = 

 = 

So we have this system of equations:

 = 
 = 

Solve that system of equations and the answer is

x = -42 and y = -60

The smaller is -60, because the SMALLER of two NEGATIVE
numbers is always the one with the LARGER absolute value.

So the answer is -60

Let's check: with the words:

-42 and -60 are in the ratio of 7 to 10 because the fraction
 reduces to  

If 24 is added to each number, we get -42+24 or -18, and
-60+24 or -36

-18 and -36 are in the ratio of 1 to 2 because the fraction
 reduces to 

Edwin