Let x be the number of chocolate cakes bought by Jeff

The total number of cakes including chocolate cakes and vanilla cakes is 288.

∴ Number of vanilla cakes = 288 - x.

Out of total cakes 4/9 are chocolate & the rest were vanilla.

∴ Number of chocolate cakes is

x = 4/9 x (total number of cakes)

= 4/9 * 288 = 1152/9 = 128

∴ Number of vanilla cakes = 288 - 128 = 160

∴ Jeff bought 128 chocolate cakes & 160 vanilla cakes

Jeff ate some chocolate cakes, the number of chocolates left was 5/8 of the total number of chocolate at first

Let y be the number of chocolate left.

∴ y = 5/8 x (total number of chocolate at first)

= 5/8 * 128 = 640/8 = 80

∴ (chocolate cake ate by Jeff) + y = Total number of chocolate cakes

∴ (Chocolate cake ate by Jeff) + 80 = 128

=> Choclate cake ate by Jeff = 128 - 80 = 48.

∴ Jeff eat 48 chocolate cakes.