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I am a bit weak at math, and I am hoping you can help me find the fastest way to solve a problem. (I hope I came to the right place). This maybe sounds ridiculous, but I want to mathematically solve this kind of problem.

I is an item in a shop.

I has a value of $500. (The price on the label)

We are the owners of this shop and we decide to apply a discount of 86% on I.

The question is: What is the minimum number of units of I that we would need to sell to have a benefit greater than or equal to what we would have had if we sold the product, without the discount?

Could you please help me solve it and explain to me, step by step, how it was calculated? I'll appreciate your help a lot.

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If I understood your problem correctly, you are originally selling only $1$ item. $\mathbf{I}$'s new price after the discount is $0.86 * 500 = 430$ dollars. The profit of selling $n$ discounted items is $ n * 430.$ Now solve $$ 430n \ge 500, $$ this gives you $n \ge 500/430 = 1.15.$ Take the ceiling (since the number of units is integer), which results in: $n = 2$ units. –  user2468 May 12 '11 at 22:47
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@sfrj: The money you get from selling an item is not the "benefit", you have to account for the cost of buying or producing the item. –  Phira May 12 '11 at 22:50
    
@M.S. It's the other way around. If the discount is 0.86, then the new price is (1-0.86)*500 –  Leonardo Fontoura May 12 '11 at 22:54
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@M.S.: A discount of 86% leaves a selling price of $70. If the cost of the unit is greater, you will never realize any benefit after the markdown. –  Ross Millikan May 12 '11 at 22:54
    
@Leonardo Fontoura Oh my bad. I didn't notice that the discount itself is $0.86$%. I thought the item is reduced to 86% of its price tag. –  user2468 May 12 '11 at 23:38
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3 Answers

up vote 5 down vote accepted

As M.S. assumed, I take it you are asking how many units of I you'd need to sell (at a discount of 86%) to equal or surpass selling just one item at $500.

Now, an 86% discount is a reduction in price in the amount of 86% of 500. Now 86% of a original price (we'll use p for price) is equal to $\frac {86}{100} \times p = 0.86 \times p$. So you plan to reduce or discount the original price of 500 by (86% of 500 $= 0.86 \times 500) = $ $430.

So, to reduce the price from $500 by 86% means you need to subtract $0.86 \times 500 = $ $430 from $500 to obtain the discounted price of I: $500 - $430 = $70.

Now, in order to equal or surpass $500 (the gross yield from selling one I at $500), we want to know (how many I) x 70 dollars is greater than or equal to 500. We let $n$ be the quantity I (how many I), with each I selling at $70.

So we have the inequality: $70 \times n \geq 500$, and now we solve for $n$. Dividing both sides of the inequality by 70 gives us $$n \geq \frac {500} {70} = \frac {50}{7} \approx 7.14.$$

Now, it makes little sense to talk about .14 of an item (say a necklace), so we round up to $n = 8$, because $7 \times 70 = 490 < 500$, whereas $8 \times 70 = 560 > 500.$

So, in this case, the answer is 8.

To generalize, if $x$ is the number of I selling for 500 dollars, and $n$ is the number of discounted I you need to sell in order to meet or surpass sales of $x$ I's sold for 500 each, then you need to have $$70 \times n \geq x \times 500$$ that is, $$ n \geq \frac {500x}{70} = \frac {50x}{7} \approx 7.14 x$$

Then if you want know how many reduced-price I you need to sell to equal or surpass the cost of any number ($x$) sold at original price, you simply substitute that value for $x$ to obtain $n$. Note: if your result after multiplying x by $\frac {50}{7}$ is not a whole number, you need to "round up" to the next integer (whole number).

If you want to take into consideration the cost of producing and marketing each I (letting c = cost per item), then you need to work with the inequality: $(70 - c)n \geq (500 - c)x$ or $$n = \frac {(500-c)x}{(70 - c)}$$

For this to mean anything, in terms of coming out with a defined and positive number $n$, the cost, c, of producing and marketing I must be such that $c < 70$.

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I understand most of it, but what confuses me, is what is the value that should replace x? Should it be replaced by the original price with no discount or by the number the seller wanted to sell with no discount? –  sfrj May 13 '11 at 8:32
    
x is for the generalized formula (following "To generalize"): it should be replaced by the number the seller wants to sell with no discount. –  amWhy May 13 '11 at 12:46
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Let...

  • Discount: d = 0.86
  • Discounted price: I' = (1 - d) * I
  • Units sold with discount: u'
  • Units sold without discount: u

We want to find...

  • u' * I' >= u * I

Thus...

  • u' * (1 - d) * I = u * I
  • u' * (1 - d) = u
  • u' = u / (1 - d)
  • u' = u / (1 - 0.86)
  • u' = u / (0.14)
  • u' = u * (50 / 7)

In other words if you sold 7 non discounted units, you'd need to sell 50 discounted units to break even.

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Let $P$ be the profit from the sale of $I$ and $C$ be the cost you, the shop owner paid for $I$. Therefore, the sale price $S$ of the product is: $S=C+P$.

Now, we'd need to know the cost of $I$, because if you give a discount that makes $S<C$ then no matter how many units of $I$ you sell you'll never make up for the fact you're selling them for less than they cost.

So, let's assume that your new price $S_d=(1-0.86)S$ is still higher than the cost $C$. Therefore, you'll need to solve $S_d*x=S$, which solving for $x$ will give you $1/1.14$, about $7.14$. Since you can't sell $.14$ units, the answer would be $8$ units to have a profit at least as high as selling one unit at $500$, assuming that your cost is lower than your discounted price.

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