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
Explanation:
Given, a : b = 2 : 5 and c: d = 5 : 2
⇒ a = 2x, b = 5x, c = 5y and d = 2y
Also, a + b + c + d = 2023 ⇒ 2x + 5x + 5y + 2y = 2023 ⇒ 7x + 7y = 2023 ⇒ x + y = 289
Now, we need to maximise a + c = 2x + 5y = 2(x + y) + 3y = 2 × 289 + 3y.
This can be maximised by maximising y (higher coefficient). Since x & y have to be natural numbers, least value of x has to be 1, hence maximum value of y will be 288.
⇒ Maximum value of a + c = 2 × 1 + 5 × 288 = 2 + 1440 = 1442
Hence, 1442.
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