The King's Reward

In your book, Maths Mela (Chapter 6, The Dairy Farm), a king wants to reward three of his most talented ministers with gold. He gives them three choices. Choice 1: take 5 gold coins and double them every day. Choice 2: take 3 gold coins and triple them every day. Choice 3: take 1 gold coin and multiply the coins by 5 every day. Minister 1 picks choice 1, Minister 2 picks choice 2, and Minister 3 picks choice 3. At the end of a week, the ministers are surprised. Who gets the most?

Let us follow the coins. Doubling means multiplying by 2, and tripling means multiplying by 3. After the coins are multiplied once, Minister 1 has 5 × 2 = 10, Minister 2 has 3 × 3 = 9, and Minister 3 has 1 × 5 = 5. Minister 3 is last!

After two times: Minister 1 has 10 × 2 = 20, Minister 2 has 9 × 3 = 27, and Minister 3 has 5 × 5 = 25. After three times: 20 × 2 = 40, 27 × 3 = 81, and 25 × 5 = 125. Now Minister 3 is ahead. After four times: 40 × 2 = 80, 81 × 3 = 243, and 125 × 5 = 625.

Multiplying by 5 makes the coins grow fastest, so every day Minister 3 pulls further ahead. By the end of the week Minister 3, who started with only 1 coin, has by far the most gold. Starting small does not matter much when you multiply by a bigger number again and again.