Math Is Fun Forum

martialtiger

Guest

Best Possible Combination

HI All,

I'm not looking for the actual answer to my question below but rather how to solve the problem.

# Problem:
##Variables
1. Costs
gold = $2
silver = $10
platinum = $30

2. Income
gold = $0.50
silver = $2.50
platinum = $8.00

So costs above is the initial cost for each item listed. As you purchase each item, the cost for the given item increases at the rate of 10% on top of the initial cost, i.e. 1st gold purchase is $2, 2nd gold purchase is $2.20, 3rd gold purchase is $2.40 and so on. Income for each item remains the same times the number of items of course, i.e. 3 gold items income is $0.50x3 ~ $1.50.

Now for the problem:
I have $100 to make purchases. How do I figure out what combination of the items I should purchase to maximize my income potential :?:

Please let me know if anything is unclear.

TIA!

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Best Possible Combination

This might not be the best way, but I'd do this by first choosing to buy no platinum, then working out what combination of gold and silver would get the most income.

Then do the same thing after buying 1 platinum and working with $70 for the rest of the calculation.
Then the same with 2 platinum, and finally again with 3 (although if you buy 3 platinum then you can't buy anything else anyway)

It's a bit fiddly, so I threw it at a computer.

Turns out the best income you can get is $24, from any of these combinations:

3P
2P 3S 1G
2P 2S 6G

It could be argued that the middle option of those is best because it costs $98, where the other two cost $99.

---

Why did the vector cross the road?
It wanted to be normal.

martialtiger

Guest

Re: Best Possible Combination

Thanks mathsyperson.  Would you know how I could construct an equation to compute this anytime?  The amount of $ to spend varies and the amount of gold, silver and platinum will be tracked continuously.