let 'x' and 'y' be the two numbers

the ratio of two numbers is 5 to 3

x : y = 5 : 3 cross multiply

3x = 5y

the sum of these numbers is 32

x + y = 31 multiply both sides by 5

5x + 5y = 155

plug 5y = 3x into the last equation

5x + 3x = 155

8x = 155

x = 155/8

y = 31 - 155/8

y = 93/8

155/8 > 93/8

the greater of the two numbers is 155/8.