If you have multiple coupons, of different discount types (amount discount vs. percent discount, that is), then there is a particular way to apply these savings that will earn more savings than the other.

*Long story short:*

__Apply the percent discount before applying the amount discount__for more savings.The concept relies on the concept of

*composition of functions*, commonly written in the form f(g(x)). These are functions that evaluate in one functions and then another, in a particular order. The topic puts emphasis on evaluating with the order of operations as well as other properties like the

*commutative property*and

*associative property*, both of which are staples in algebraic concepts.

To help solidify their learning, or ward off skeptics on the topic, we try a couple examples: a $50 purchase and a $100 purchase, for ease of computation. Follow along:

__For a $50 purchase:__$50 - $10 discount = $40, then $40 - (0.2)*$40 =__$10 discount applied first:____$32 final price__: $50 - (0.2)*$50 = $40, then $40 - $10 =__20% discount applied first____$30 final price__

__For a $100 purchase:__$100 - $10 discount = $90, then $90 - (0.2)*$90 =__$10 discount applied first:____$72 final price__- 20% discount applied first: $100 - (0.2)*$100 = $80, then $80 - $70 =
__$70 final price__

It is no strong coincidence that there is a two-dollar difference between these scenarios. This is the critical piece of what I'm trying to convey with this post (oh, and it's a fun math lesson to teach).

In using the dollar-discount first, the consumer or retailer is causing the percent-discount to be applied to a smaller value, therefore not letting it stretch as far as it could otherwise. It is, in essence, as if you are also taking the percent OFF of the dollar discount you wish to use (note the examples above, where there is a $2 difference and $2 is 20% of $10).

In applying the percent discount first, your percent covers the larger original purchase price and stretches further. This is the gist of why the percent coupon should be applied first, when you have the opportunity.

*NOTE: Some retailers have a point-of-sale system which is programmed to use the dollar discount first, no matter when you had a coupon to the cashier, so you might be forced to play their game and sacrifice part of your savings. I will not name names, but have had a couple of disappointed students bring in receipts and show me how much MORE they could have saved if the register didn't force the order of discounts applied.*

I'm making a bunch of links in this sentence if you would like either the

*TI-Nspire file*I have shown up above in the slideshow, or the original

*presentation*and handout I did for this topic to earn

*T^3 Instructor*status a few summers ago. (Don't have a TI-Nspire? Try running the TI-Nspire file through the

*TI-Nspire Document Player*without a need for download of software.)

*curious sale display*, pictured at left. To use the word "PLUS" in this instance seemed

*additionally*confusing (yes, pun intended there), and caught the ire of Dan Meyer on

*his blog*shortly after I'd posted the photo to

*101qs.com*.

This, and other sale opportunities like it have been a common cause of commotion in class among my students who have heard me teach this lesson. They are able to bring up new shopping adventures they have had, where they might have corrected the way a cashier had rang something up, or a discount was applied more favorably than they expected, or the discount merely balanced out the sales tax they would have paid anyway. Regardless, the lesson sticks.

Applying similar, consecutive discounts have proven to be a decent introduction to exponential growth, since students are aware that "50% off, then another 50% off does not make the item FREE." For those students who are very shopping-savvy, these applications hit home for them much more quickly than any compound interest problem ever could.

Speaking of interest, I hope this sparked some of yours. Have a good day and enjoy the holiday season, saving all the while!

--Keltner--