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This may seem like a homework problem because it is. However it is not my homework - it belongs to the child I am tutoring, so please feel free to give a full answer, as I will only lead the child along with it and not just fill the answers for him.

The question:

Two stores offer discounts on a particular class of products (let's say laptops). One store offers a discount of 30% off for every hundred dollars you spend. The other offers a 20% discount on your total.

Where should you buy your laptop?

The answer is obviously it depends. Clearly, for prices in even multiples of hundreds, the first offer (30% off every hundred) is far better, but for steps of 100 in \$99 (\$99, \$199, \$299 ....), the second offer is better. And they are equal at steps of 50, as in \$150, \$250, etc.

But how do you represent all this as a mathematical formula? How can you mathematically solve which is better?

I suppose the equations would look something like

$$\frac{7}{10}x + y, \space \space \space \space \space \space \frac{4}{5}(x+y)$$

where $x$ is the multiples of $100$ and $y$ is the number of $1$s. But where do you go from here?

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After a (short) while the first is always better. A single formula is of limited usefulness. – André Nicolas May 31 '14 at 19:23
@ Andre, for practical price ranges of a laptop, total discount is better all the time. I am trying to find the breakeven ( you call it flip between choices) within this practical range, but could not find it – satish ramanathan May 31 '14 at 19:51
In symbols, let the price be between $100n$ (inclusive) and $100(n+1)$ (exclusive). Discount 1 is $30n$, Discount 2 is $\lt 20(n+1)$. So 1 is better than 2 if $30n\ge 20(n+1)$, that is if $n\ge 2$. So only issue is with $[100,200)$, easy. – André Nicolas May 31 '14 at 20:15


Draw out the equation for a price range from 100 to 4000 in increments of 100 and workout the discounted Price, you will find that the 20% discount on the total would be worse than the 30% on every 100 dollars. Chart out the equations in EXCEL and calculate for different prices and compare them. You will find the result self evident as you see below.

enter image description here

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The After Discount 1 column does not look right. Mental calculation is more reliable. – André Nicolas May 31 '14 at 19:56
I edited it and the flip point happens right at $200. Andre cannot be wrong!!. Thanks for noticing. – satish ramanathan May 31 '14 at 19:57

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