superprismatic Posted May 11, 2011 Report Share Posted May 11, 2011 Shyster Al owns a furniture store. He's always looking for a way to lure customers away from his competitors. But, in the cut-throat furniture business, the best deal a dealer can give to a customer is a 25% discount below the MSRP (Manufacturer's Suggested Retail Price) and still make the necessary profit to keep the store viable. One fine day, Al hit on a beautiful scheme for his SAFE (Shyster Al's Furniture Emporium) store which makes it seem as though his customers can get something close to a 33.333...% discount. His scheme is this: If a customer buys several items, he pays full MSRP price for the first item. For subsequent items, he pays full price minus 1/3 of what he had to pay after the discount due on the previous item. For example, if a customer is buying 4 items with these MSRPs, in order, $600, $650, $900, and $300. He pays full price for the first item but gets a discount of $200 on the second item (1/3 of $600) for which he pays $450. This gives him a $150 discount on the third item (1/3 of $450) which costs him $750. So, 1/3 of his net cost of the third item ($250) will be his discount on the fourth, which will now only cost his another $50. So, he pays $1850 ($600+$450+750+$50) for $2450 MSRP of furniture. Each discount is limited by the price of the item being bought, so, e.g., a discount of $100 on something costing $50 will revert to a $50 discount. Also, all amounts are rounded to the nearest cent. Now, you wish to minimize what you will pay for 10 pieces of furniture with MSRPs of $487, $403, $315, $289, $262, $252, $233, $229, $208, and $198. In which order should you place these items so as to minimize your cost? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted May 12, 2011 Report Share Posted May 12, 2011 487,198,403,208,315,229,289,233,262,252 hope thats right Quote Link to comment Share on other sites More sharing options...
0 Guest Posted May 12, 2011 Report Share Posted May 12, 2011 The order to buy is 262, 252, 289, 233, 315, 229, 403, 208, 487, 198 Total overall discount will be 24.6% I solved it by realizing that the price can be written this way, with x1,x2,x3...being the msrp for each item total = x1 + x2 - x1/3 + x3 - (x2-x1/3)/3 + x4 - (x3-(x2-x1/3)/3)/3 and so on. Just looking at the discount portions and then simplifying, discount for item n = xn-1/3 + xn-2/32 - xn-3/33... What this means is the first item will give a 1/3 discount to the next, but will add 1/9 to the next after that, then deduct 1/27 to the one after, and so on through to the last item. This totals about 1/4 spread out over the entire purchase with the exception of the last 3 items. The last item gives no discount, the 9th item gives 1/3 discount, and the 8th item gives a total of 22.22% off. So the cheapest item should be bought last, the most expensive bought 9th, and the 2nd cheapest bought 8th. For the other 7 items, I sorted in order to put the more expensive items in the higher discount order. The order of items that give the best discount is item 9,7,5,3,1,2,4,6,8,10 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted May 12, 2011 Report Share Posted May 12, 2011 The prices will have to be like Max prices in decreasing order with in beween the lowest prices in ascending order... i.e. $487,$198, $403, $208, $315, $229, $289, $233, $262, $252 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted May 12, 2011 Report Share Posted May 12, 2011 Due to rounding there are two orderings that will result in the maximum discount available of 24.55% for a total purchase cost of $2170 for the $2876 valued items. (1) $262, $252, $289, $233, $315, $229, $403, $208, $487, $198 and (2) $289, $252, $262, $233, $315, $229, $403, $208, $487, $198 Without intermediate rounding, the ordering (1) will give the greatest discount of 24.60% for a total purchase cost of $2168.40. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted May 12, 2011 Report Share Posted May 12, 2011 Using a computer program, with intermediate rounding, the following 10 orderings gives the maximum discount of 24.58275% for a total purchase cost of $2169: ( 1) $315, $262, $229, $233, $289, $252, $403, $208, $487, $198 ( 2) $289, $252, $262, $229, $233, $315, $403, $208, $487, $198 ( 3) $289, $252, $229, $315, $262, $233, $403, $208, $487, $198 ( 4) $262, $252, $289, $315, $229, $233, $403, $208, $487, $198 ( 5) $252, $315, $233, $289, $262, $208, $403, $229, $487, $198 ( 6) $252, $315, $233, $262, $289, $229, $403, $208, $487, $198 ( 7) $252, $289, $233, $262, $315, $229, $403, $208, $487, $198 ( 8) $252, $262, $315, $289, $233, $229, $403, $208, $487, $198 ( 9) $252, $229, $289, $233, $315, $262, $403, $208, $487, $198 (10) $229, $289, $252, $315, $262, $233, $403, $208, $487, $198 Without intermediate rounding the ordering with the lowest purchase cost of $2168.40 is: $262, $252, $289, $233, $315, $229, $403, $208, $487, $198 And the order of items by its MSRP -- ranked 1=lowest, 10=highest -- would be: 6, 5, 7, 4, 8, 3, 9, 2, 10, 1 Quote Link to comment Share on other sites More sharing options...
0 Peekay Posted May 12, 2011 Report Share Posted May 12, 2011 $262, $252, $289, $233, $315, $229, $403, $208, $487, $198 giving a total discount of $707.61 with rounding of the final prices but not the discounts. Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted May 14, 2011 Author Report Share Posted May 14, 2011 Nana7, Dej Mar, and Peekay got it correct to within some small rounding differences. Nice job (and quick)! Quote Link to comment Share on other sites More sharing options...
Question
superprismatic
Shyster Al owns a furniture store. He's always looking for
a way to lure customers away from his competitors. But, in
the cut-throat furniture business, the best deal a dealer can
give to a customer is a 25% discount below the MSRP (Manufacturer's
Suggested Retail Price) and still make the necessary profit
to keep the store viable. One fine day, Al hit on a beautiful
scheme for his SAFE (Shyster Al's Furniture Emporium) store
which makes it seem as though his customers can get something
close to a 33.333...% discount. His scheme is this: If a
customer buys several items, he pays full MSRP price for the
first item. For subsequent items, he pays full price minus
1/3 of what he had to pay after the discount due on the previous
item. For example, if a customer is buying 4 items with these
MSRPs, in order, $600, $650, $900, and $300. He pays full price
for the first item but gets a discount of $200 on the second
item (1/3 of $600) for which he pays $450. This gives him a $150
discount on the third item (1/3 of $450) which costs him $750.
So, 1/3 of his net cost of the third item ($250) will be his
discount on the fourth, which will now only cost his another $50.
So, he pays $1850 ($600+$450+750+$50) for $2450 MSRP of furniture.
Each discount is limited by the price of the item being bought,
so, e.g., a discount of $100 on something costing $50 will revert
to a $50 discount. Also, all amounts are rounded to the nearest
cent.
Now, you wish to minimize what you will pay for 10 pieces of
furniture with MSRPs of $487, $403, $315, $289, $262, $252, $233,
$229, $208, and $198. In which order should you place these items
so as to minimize your cost?
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