Kelvin and Patrick had some money in the ratio 3:5. The ratio of Kelvin's expenditure was 3:4. If Kelvin spent 1/3 of his money and Patrick had $55 left, how much money did Kelvin have at first?

kelvin had 3x and spent 3y

patrick had 5x and spent 4y
3y = 1/3 (3x)
3y = x
5x-4y = 55
11y = 55
y = 5
x = 15
so kelvin had 3*15 = $45 to start
check:
kelvin had 45 and spent 15
patrick had 75 and spent 20, leaving 55