Mental Math: combining discounts and bonus quantities

The other day I found a great deal.

A snack I like was now being packaged with 10 percent more food, AND it was being sold at a 20 percent discount.

I had no hesitation in buying since it was clear I was getting a good price.

But later I was curious how good of a deal I was getting.

How much was I saving? To be more precise, how much less was the unit price?

I came across a simple formula that helped me figure this out.

Your instincts are pretty close

Your first guess may have been to add the bonus quantity of 20 percent to the discount of 10 percent, to give a total of 30 percent off.

This is clearly wrong mathematically, as bonus quantities and discounts do not add up.

And yet, if you would have made this guess, it would not be very far off.

Adding the bonus quantity and the discount percentage turns out to be fairly accurate.

The exact formula

We can derive the exact formula as a starting point.

Without loss of generality, suppose the original product cost $1 per kilogram.

The item on sale would offer a 10 percent increase to 1.1 kilograms, and it would sell at $0.80 at a 10 percent discount. The unit price would be $0.73 per kilogram.

This product would offer a 1 – 0.73 / 1.00 = 27 percent discount.

In more general terms, if a product offered a bonus quantity B, and a cost discount of C, the the resulting unit price savings would be:

1 – (1 – C) / (1 + B)

It turns out there is a way we can simplify this formula for certain ranges of bonus quantities.

An approximating formula

We can simplify the exact formula as follows:

1 – (1 – C) / (1 + B) ~ 1 – (1 – C)(1 – B),
using the first-order Taylor series approximation of 1/(1 + B)

We can further simplify to get the result:

1 – (1 – C)(1 – B) = C + B – CB ~ C + B,
using another approximation that CB is close to 0

So all you really need to do is add up the cost savings and the bonus quantity to get a rough idea of your savings on the unit cost.

In other words,

Unit cost savings ~ Cost discount + Bonus quantity percentage

In my example of a 10 percent bonus discount, and a 20 percent cost savings:

–the exact formula gives 27.3%

–approximating formula gives 30%

While the approximating formula is off on the order of 10 percent, this is probably good enough for mental math when taking shopping trips at your local supermarket.