The sum of two numbers is 51. if one fourth of one number exceeds one fifth of another number by 6, which of the following is the smaller number?

Let x = 1st number and y = 2nd number
Using the sentences:
The sum of two numbers is 51:  x + y = 51
one fourth of one number exceeds one fifth of another number by 6:
(1/4)x > (1/5)y + 6

This means y is the smaller number.
From x + y = 51 → x = 51 - y
Plug into the inequality
(1/4)(51 - y) > (1/5) y + 6
Multiply each side by 20
5(51 - y) > 4y + 120
251 - 5y > 4y + 120 
-9y > -131
y < 

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