Question 11 - Figure it out - Page 145-147 - Chapter 6 Class 8 - Algebra Play (Ganita Prakash II)

Transcript

Question 11 Karim and the Genie Karim was taking a nap under a tree. He had a dream about a magical lamp and a genie. He heard a voice saying, “I have come to serve you, Oh master”. He woke up and to his surprise, it was a genie! “Do you want to make money?”, asked the genie. Karim nodded dumbly in bewilderment. The genie continued, “Do you see the banyan tree over there? All you have to do is go around it once. The money in your pocket will double”. Karim immediately started towards the tree, only to be stopped by the genie. “One moment!”, said the genie. “Since I am bringing you great riches, you should share some of your gains with me. You must give me 8 coins each time you go around the tree.” Thinking that was a trifling amount, Karim readily agreed. He went around the tree once. Just as the genie had said, the number of coins in his pocket doubled! He gave 8 coins to the genie. He made another round. Again the number of coins doubled. He gave 8 more coins to the genie. He went around the tree for the third time. The number of coins doubled again, but to his horror, he was left with only 8 coins, exactly the number of coins he owed the genie! As Karim began to wonder how the genie tricked him, the genie let out a loud laugh and disappeared. (i) How many coins did Karim initially have?Let Karim's initial number of coins = 𝒙. The rule for every single round is: Double the current coins, then subtract 8. Round 1 Start with: 𝑥 Double it: 2𝑥 Amount after paying the genie: 𝟐𝒙−𝟖 (This is how much money Karim has in his pocket at the end of Round 1). Round 2 Start with: 2𝑥−8 Double it: 2(2𝑥−8) Let's expand this: 4𝑥−16 Amount after paying the genie : (4𝑥−16)−8=4𝑥−24 (This is how much money he has at the end of Round 2). Round 3 Start with: 4𝑥−24 Double it: 2(4𝑥−24) = 8𝑥−48 Now, the book says that after the coins doubled this third time, Karim looked and saw he was "left with only 8 coins, exactly the number of coins he owed the genie!". Thus, Amount after Round 3 = 8 𝟖𝒙−𝟒𝟖=𝟖 𝟖𝒙−𝟒𝟖=𝟖 8𝑥=8+48 8𝑥=56 𝑥=56/8 𝒙=𝟕 So, Karim started with exactly 7 coins. Question 11 (ii) For what cost per round should Karim agree to the deal, if he wants to increase the number of coins he has?We now know Karim's starting coins = 7. Let the Genie's new fee = y Let’s play First Round Start with: 7 Double it: 2 × 7=14 Amount after paying genie: 𝟏𝟒−𝒚 (This is how much money Karim has in his pocket at the end of Round 1). For Karim to make money, his ending amount (14 – y) must be greater than (>) his starting amount (7) So, our equation becomes 14 – y > 7 – y > 7 – 14 – y > –14 + 7 – y > –(14 – 7) – y > –7 Multiplying by –1 both sides If we multiply by negative number, the sign reverses So, inequality becomes –y × –1 < –7 × –1 y < 7 Therefore, to actually make a profit, Karim must agree to a cost of 6 coins or fewer per round. Question 11 (iii) Through its magical powers, the genie knows the number of coins that Karim has. How should the genie set the cost per round so that it gets all of Karim’s coins?Let's prove it with variables so it works for any starting amount. Let Karim's starting coins =𝒄 Let the genie's fee =𝒈 For the genie to slowly drain Karim's pockets, Karim's money after Round 1 must be strictly less than (<) what he started with. The formula for Round 1 is: 𝟐𝒄−𝒈. So, we set up the inequality: 𝟐𝒄−𝒈<𝒄 Solving the Inequality: Subtract 2𝑐 from both sides to get the 𝑔 by itself: −𝑔<𝑐−2𝑐 −𝒈<−𝒄 Multiply both sides by -1 (and flip the sign!): 𝒈>𝒄 The algebra shows us exactly how the trick works. To drain a human of all their money, the genie simply needs to set his fee (𝑔) to be any number greater than the human's starting coins (c). Since Karim had 7 coins, a fee of 8 guaranteed his doom! Solving the Inequality: Subtract 2𝑐 from both sides to get the 𝑔 by itself: −𝑔<𝑐−2𝑐 −𝒈<−𝒄 Multiply both sides by -1 (and flip the sign!): 𝒈>𝒄 The algebra shows us exactly how the trick works. To drain a human of all their money, the genie simply needs to set his fee (𝑔) to be any number greater than the human's starting coins (c). Since Karim had 7 coins, a fee of 8 guaranteed his doom!