# Mathematically choose the better discount

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.

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