Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum possible value of a + c is________.

Let us assume the proportionaliuty constant for the first ratio as  and the second ratio as t.

So, a = 2k and b = 5k. And  c = 5t and d = 2t

a+b+c+d = 7k+7t = 2023

k+t = 289

We need to maximise 2k+5t. The increase in t will contribute more to the value of 2k+5t. But at the same time k can;t be 0. So, k has to be at least 1. 

If k = 1, t = 288.

2k+5t = 2+288*5 = 1442