Answer to Riddle #24: The Market

24. You're a farmer. You're going to a market to buy some animals. On the market there are 3 types of animals for sale. You can buy: Horses for £10 each, goats for £1 each and ducks, you get 8 of these per bunch and each bunch costs £1. The aim is to acquire 100 animals at the cost of £100, what is the combination of horses, goats and duck that allows you to do this? (you must buy at least one of each.)

This puzzle was e-mailed to me by Shaun Ball, for the record this one took me about 20 minutes to solve by brute force, how long did it take you...?

picture of a horse, sheep and duckBefore reading the answer can I interest you in a clue?

This is problem can be solved relatively easily by using a Brute Force technique, that is to say trying lots of combinations. There are some things you can do to speed up your working....
  • We can have a maximum of 10 horses or we would have no money left.
  • We can have a maximum of 12 bunches of ducks or we would have too many animals.
  • A spread sheet might help
However there is a more eloquent approach (thanks to Gary Short):

This can be solved as a simultaneous equation, with three variables and two equations...

Our equations are:
  1. 10*H + G + D/8 = 100 (pounds)
  2. H + G + D = 100 (animals)
Subtracting equation #2 from equation #1 gives:
9H - 7D/8 = 0
OR....
9H = 7D/8
(at this point we can actually stop solving the equation as we know that ducks come in bunches of 8 so the feature D/8 in the equation actually represents the number of bunches of duck, therefore the equation is simply 9H = 7DB which easily gives us a solution. But we will continue....)

72H = 7D
Directly this equation is unsolvable as there are an infinite number of solutions. However, we have the restraint that both variables must be integers and that there H<100 & D<100.

Hence the only solutions is
H = 7 : D = 72 : G = 21

The answer is 7 horses, 21 goats and 9 bunches of ducks.







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