1. While on a trek to walk into Mordor, you were captured by the minions of Saur

Question

1. While on a trek to walk into Mordor, you were captured by the minions of Sauron and taken to Mount Doom. He had heard of your jewel making ability and is enslaving you to create the Rings of Power. He instructs you that you need to create nineteen identical rings, three for elves, seven for dwarves and nine for humans. In addition to this, you need to create one ring to rule them all. He informs you that they are to be created from pure gold and you are to advise him an appro- priate proportion of his power that should be put into each ring to ensure they remain under control of his ring that rules them all. The rings are also to be created within the following guidelines. ? A total of 168g of gold is available for all of the rings. ? Sauron desires to put 87% of his power into all of the rings. ? There must be twice as much gold in the one ring to rule them all as the other nineteen. ? The percentage of Sauron's power put into the one ring to rule them all must be 10 times greater than the other nineteen rings. (a) Formulate this problem as a system of linear equations. Hints: For example, let gi be the number of grams of gold used and pi be the percent- age of Sauron's power in the one ring to rule them all. You can use the relationship (number of items) x (amount per item) to determine the total amount of an item used. (b) Solve the system of linear equations from (a), showing all working. (c) Interpret the solution obtained from (b) and advise Sauron how you plan to create his rings.

Explanation / Answer

1. Gold weight:

   Let g1=number of grams of gold in Sauron's ring to rule them all

   Then, g2=number of grams of gold in each of the other 19 rings

   According to problem=> g1 + 19*g2 = 168 ----------------eqn-1

   Also, given that => g1=2*g2 ------------------eqn-2

   So, from above equations=>2*g2 + 19*g2 = 168=>g2=168/21=8 grams.

   Hence, g1=2*g2=2*8=16 grams.

2. Sauron's power percentage:

Let p1=percentage of Sauron's power in the ring to rule them all

   Then, p2=percentage of Sauron's power in each of the other 19 rings

   Then according to problem, p1 + 19*p2 = 87 -------------------------eqn-3

   Also given that => p1=10*p2 --------------------------eqn-4

   So, from above equations=>10*p2 + 19*p2 = 87 => p2=87/29=3 percent

   Also, p1=10*p2=10*3=30 percent

**NOTE=>Part (a) and (b) have been solved in above solutions.

Part (c)

For explanation to Sauron => I will create your ring to rule them all, by using 16 grams gold

and it will be having 30 percent of your power in it.

I will create the other 19 rings of same type, with each having 8 grams gold and your 3 percent

of power in it.

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