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### Shopping Math Or, How To Convince Yourself To Buy Anything You Want

Lord and Taylor department store is playing puzzles with us this Holiday season. An ad on the window reads: "The right dress is worth it's weight in gold or silver" with the "right" light and sparkling T-shirt dress available in gold or silver colors. A 100 or so gram dress for \$108. Very clever! In Manhattan, a giant SALE signs are posted in every store window as a reminder of the past holidays and the ongoing recession. Some of the stores are even offering double-discounts: 30% off on top of the previously announced 50% off, or 50% off all the red sticker items that already have been discounted by 30%. At the register, many of us nervously wonder whether the order of discounts matters. Would a 30% discount on top of 50% off be the same as 50% off on top of a 30% discount? Why wouldn't they just announce an 80% discount? We shouldn't allow ourselves to be fooled with this. With numbers so attractively appealing, lets look into this closely.

If a dress costs \$100, after the initial 50% off it will be (0.5 x \$100) = \$50. After the secondary 30% off, it will be 100%-30%=70% of the previously discounted price: (0.7 x (0.5 x \$100)), that is (0.35 x \$100) = \$35. We will end up paying only 35% of the price, having a 100%-35% = 65% discount. In the second scenario, the initial 30% discount would result in a 100% - 30% = 70% dress price of (0.7 x \$100) = \$70, and the additional 50% off will produce (0.5 x (0.7 x \$100)) , that is (0.35 x \$100) = \$35. Same as the first scenario. So, discount order does not matter. But 30% and 50% would not give us 80%, because they are applied one on top of the other, producing multiplication and not an addition. Discounts of 30% and 50% would result in a 65% discount, which is not quite 80% but is still pretty good.

What if, even after a discount, the price still has more zeros than you would like to imagine spending? But the item looks too good to pass by. Try calculating the cost-per-wear price. Suddenly, a nice \$50 pair of pants that would be worn only once a year becomes more expensive than a fancy \$300 pair of jeans that will be covering your lower body more often. Assuming a 3 year pants' lifespan, the cheaper pants would have the following cost-per-wear: \$50/(1 time per year x 3 years) = around \$17 per-wear. The more expensive jeans that can be worn at least once a week may have a surprisingly lower cost-per-wear: \$300/(52 times per year x 3 years) = less than a \$2 per wear. Makes you see things in a whole new light, doesn’t it.

Unbeknownst to most of us, shopping involves very complex math. In a matter of minutes our brain takes into account many independent variables: uniqueness of the item, fit, color, fashion trends, price, salary, how much you've already spent this month, probability of finding the same item in the discount store, chance of friends or coworkers having this item, next sale, your free time, and gas prices. Also, in addition to retail price, discounted price, and cost-per-wear price, we can come up with a multitude of different price to value strategies. For example, when considering cost-per-time spent shopping, one could suggest buying the first matching item we see instead of searching for a cheaper one for hours or months.

Cost is also subjective. Some of us turn to online retailers because paying shipping charges feels less costly than a stressful day at the mall with family or the guilt for not being with them on Sunday evening. Others insist on real in-store shopping because window displays, browsing and trying items on bring so much joy, no time savings can beat that. We can also consider a cost-per-pleasure-of-wearing price that can help many people justify buying super-expensive brand name items that come with a proudly worn status tag.

Keeping all this in mind, it should be easy to find an appropriate cost parameter to convince yourself to buy almost anything you want, or to talk yourself (or your spouse) out of any purchase.

Try something else from The Math Mom: Cheap And Hip But Assembled By You.

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She can't be fooled anymore :-)

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