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K Sengupta
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Item purchase and bargain puzzle
« on: Jan 22nd, 2010, 7:42pm » |
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Saturday, Kelly and four of her friends, lured by advertisements of the "biggest sale of the year," went Christmas shopping at Tracy's. Each of the five found the sales pitch to be true, finding a gift for her dad--one buying a sweater--at a great price.
Given the clues below, determine the item each purchased, its original price, and its price on sale Saturday.
(i) ) The girls ended up spending $134 total on their five gifts, from a low of $21 to a high of $36,
(ii) The original price of Lynne's gift was $50 more than the original price of the gloves another girl bought her dad, but after Tracy's reductions Lynne spent only $8 more than the gloves purchaser did.
(iii) The price of Jada's gift was reduced 20% more than the price of the tie, but Jada still paid $4 more for her item than the girl who bought the tie for her dad paid.
(iv) The original prices ranged from a low of $30 to a high of $120, totaling $310 for the five gifts.
(v) Maria saved $7 more than the girl who bought the belt did.
(vi) The sale price of the dress shirt was $4 more than the sale price of the item Nicole purchased.
(vii) The biggest discount on any of the five items was 70% and the smallest discount was 30%.
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Ant_Man
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Re: Item purchase and bargain puzzle
« Reply #1 on: Jan 23rd, 2010, 3:27pm » |
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I think clue III might be confusing. We apply Tracy's discount to Lynne? Do we take the Tracy's percent discount from Lynne's or the number of dollars that Tracy saved?
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rmsgrey
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Re: Item purchase and bargain puzzle
« Reply #2 on: Jan 25th, 2010, 6:51am » |
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on Jan 23rd, 2010, 3:27pm, Ant_Man wrote:| I think clue III might be confusing. We apply Tracy's discount to Lynne? Do we take the Tracy's percent discount from Lynne's or the number of dollars that Tracy saved? |
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Tracy's is the name of the shop, so Tracy's reductions apply to all items in the sale, rather than being the reduction associated with a single shopper...
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #3 on: Jan 25th, 2010, 7:29am » |
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on Jan 23rd, 2010, 3:27pm, Ant_Man wrote:| I think clue III might be confusing. We apply Tracy's discount to Lynne? Do we take the Tracy's percent discount from Lynne's or the number of dollars that Tracy saved? |
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To give examples of possible meanings:
1. Tie was reduced 10%, Jada's gift was reduced by 30%, difference = 20%
2. Tie was reduced by $10, Jada's gift was reduced by $12, and 12 is 20% greater than 10
Which, if either, of those is correct, please?
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #4 on: Jan 25th, 2010, 8:33am » |
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on Jan 23rd, 2010, 3:27pm, Ant_Man wrote:| I think clue III might be confusing. We apply Tracy's discount to Lynne? Do we take the Tracy's percent discount from Lynne's or the number of dollars that Tracy saved? |
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Actually, even without that clue, there doesn't seem to be a feasible solution. Could the person who set the puzzle double check the clues, please?
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #5 on: Jan 25th, 2010, 12:02pm » |
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on Jan 25th, 2010, 8:33am, MathsForFun wrote:| Actually, even without that clue, there doesn't seem to be a feasible solution. Could the person who set the puzzle double check the clues, please? |
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Here's the closest that it is possible to get: this solution fails because the most expensive original item is $112.0625 instead of $120, per clue 4:
Jada: BUYER:gloves ORIGINAL:$62.0625 SALE:$28
Kelly: BUYER:tie ORIGINAL:$64.875 SALE:$24
Lynne: BUYER:sweater ORIGINAL:$112.0625 SALE:$36
Maria: BUYER:shirt ORIGINAL:$41 SALE:$25
Nicole: BUYER:belt ORIGINAL:$30 SALE:$21
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #6 on: Jan 25th, 2010, 12:17pm » |
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on Jan 23rd, 2010, 3:27pm, Ant_Man wrote:| I think clue III might be confusing. We apply Tracy's discount to Lynne? Do we take the Tracy's percent discount from Lynne's or the number of dollars that Tracy saved? |
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If I ignore that non-understandable clue, then I am able to produce the following solution for the other clues:
Jada: GIFT:gloves ORIGINAL:$70 SALE:$28
Kelly: GIFT:tie ORIGINAL:$49 SALE:$24
Lynne: GIFT:sweater ORIGINAL:$120 SALE:$36
Maria: GIFT:shirt ORIGINAL:$41 SALE:$25
Nicole: GIFT:belt ORIGINAL:$30 SALE:$21
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rmsgrey
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Re: Item purchase and bargain puzzle
« Reply #7 on: Jan 26th, 2010, 4:41am » |
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on Jan 25th, 2010, 12:17pm, MathsForFun wrote:
If I ignore that non-understandable clue, then I am able to produce the following solution for the other clues:
Jada: GIFT:gloves ORIGINAL:$70 SALE:$28
Kelly: GIFT:tie ORIGINAL:$49 SALE:$24
Lynne: GIFT:sweater ORIGINAL:$120 SALE:$36
Maria: GIFT:shirt ORIGINAL:$41 SALE:$25
Nicole: GIFT:belt ORIGINAL:$30 SALE:$21 |
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Modifying that solution, I get:
Jada: GIFT:gloves ORIGINAL:$70 SALE:$28
Maria: GIFT:tie ORIGINAL:$40 SALE:$24
Lynne: GIFT:sweater ORIGINAL:$120 SALE:$36
Kelly: GIFT:shirt ORIGINAL:$50 SALE:$25
Nicole: GIFT:belt ORIGINAL:$30 SALE:$21
which makes the percentage discount on Jada's gift 20 more than that on the tie, and makes the percentage discounts and original prices all plausibly round numbers.
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #8 on: Jan 26th, 2010, 7:08am » |
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on Jan 26th, 2010, 4:41am, rmsgrey wrote:| Modifying that solution, I get {snip} which makes the percentage discount on Jada's gift 20 more than that on the tie, and makes the percentage discounts and original prices all plausibly round numbers. |
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Aha! I was closer than I had realised! All I needed was to know was how clue 3 works, then take the final step:
In my solution, the gloves had a 60% discount - so 70*0.4=28. All I needed to do was to apply a 40% discount to the tie - so the original price becomes (5/3)*24 which equals 40 - and then adjust the original price of the shirt up by the $9 taken from the tie.
For future reference, clue 3 should read as follows:
(iii) The percentage reduction on Jada's gift was 20 more than the percentage reduction on the tie, but Jada still paid $4 more for her item than the girl who bought the tie for her dad paid.
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MathsForFun
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Re: Item purchase and bargain puzzle
« Reply #9 on: Jan 26th, 2010, 3:15pm » |
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on Jan 26th, 2010, 4:41am, rmsgrey wrote:| ...and makes the percentage discounts and original prices all plausibly round numbers. |
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I think that if you know that the original and sale prices are all round numbers, then this puzzle is actually well within reach of brute force: one price range has 16 possible values, the other, 91. Of these, two are fixed, 2 can vary, and one is calculated from the other 4.
If every condition that can be is checked before each deepening of the nested loops, I suspect that such a program could complete very quickly.
The method I used was a combination of linear programming and trial and error. Once my GNU Mathprog model was complete, it got me close to the solution without me having to do much experimentation. Will post the final model if anyone is interested.
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