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...?
Before 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:
10*H + G + D/8 = 100 (pounds)
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.
Both of them got this wrong.
If you're curious what Bard made of this puzzle...
The combination of horses, goats, and ducks that allows you to acquire 100 animals at the cost of £100 is:
9 horses
91 goats
1 duck
This combination costs £90 + £91 + £1 = £182. However, you are allowed to spend up to £100, so you can return 82 animals to the market. This will leave you with 100 animals at the cost of £100.
Here are the steps on how to acquire 100 animals at the cost of £100:
Buy 9 horses for £90.
Buy 91 goats for £91.
Buy 1 duck for £1.
Return 82 goats to the market.
You will now have 100 animals at the cost of £100.
If you're curious what ChatGPT made of this puzzle...
The below answer IS WRONG
Let's start with the minimum number of animals we need to buy from each type, which is one.
We have to buy a total of 100 animals, so we need to buy 99 more animals.
Let's assume we buy x horses, y goats, and z bunches of ducks.
The total cost of buying x horses is 10x.
The total cost of buying y goats is y.
The total cost of buying z bunches of ducks is z.
We know that the total cost of buying all these animals is £100. So we can write an equation:
10x + y + z = 100
We also know that we need to buy 100 animals in total, so:
x + y + 8z = 100
Now we have two equations with two variables, which we can solve to find the values of x, y, and z that satisfy both equations.
First, we can rearrange the second equation to get:
x + y = 100 - 8z
We can substitute this expression for x + y into the first equation:
10(x + y) + z = 100
Substituting the expression we just found:
10(100 - 8z) + z = 100
Expanding:
1000 - 80z + z = 100
Simplifying:
920 = 79z
So:
z = 920/79 ˜ 11.65
Since we must buy an integer number of bunches of ducks, we'll round this up to 12. This means we need to buy 12 bunches of ducks, which will give us 12 x 8 = 96 ducks.
Now we can go back to the second equation:
x + y + 8z = 100
Substituting z = 12:
x + y + 8(12) = 100
x + y = 4
We need to find two positive integers x and y that add up to 4. The only possibility is x = 1 and y = 3.
So we need to buy:
1 horse for £10
3 goats for £3
12 bunches of ducks for £12
This gives us a total of 1 + 3 + 12 x 8 = 97 animals, which is one short of our goal of 100. We still have £1 left to spend, so we can buy one more goat to reach our goal:
1 horse for £10
4 goats for £4
12 bunches of ducks for £12
This gives us a total of 1 + 4 + 12 x 8 = 100 animals for a total cost of £26.
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