Dudeney, Amusements in Mathematics Problem 3 Three countrymen met at a cattle market. "Look here, " said Hodge to Jakes, "I'll give you six of my pigs for one of your horses, and then you'll have twice as many animals here as I've got." "If that's your way of doing business," said Durrant to Hodge, "I'll give you fourteen of my sheep for a horse, and then you'll have three times as many animals as I." "Well, I'll go better than that, " said Jakes to Durrant; "I'll give you four cows for a horse, and then you'll have six times as many animals as I've got here." No doubt this was a very primitive way of bartering animals, but it is an interesting little puzzle to discover just how many animals Jakes, Hodge and Durrant must have taken to the cattle market. Solution
: Jakes 7, Hodge 11, Durrant 21, total 39. Let J, H, D be the number of animals brought by Jake, Hodge, Durrant respectively. Hodge offers to transfer a net 5 animals to Jake. So J + 5 = 2(H - 5). Similarly, 3(D - 13) = H + 13, and 6(J - 3) = D + 3. So: 2H - J = 15 (1) 3D - H = 52 (2) -D + 6J = 21 (3) (1) + 2 (2) gives 6D - J = 119. Adding 6 (3) gives 35J = 245, so J = 7. Then (1) gives H = 11 and (3) gives D = 21. Dudeney 1-25 (C) John Scholes jscholes@kalva.demon.co.uk 25 Oct 2002