Here is a real-life example that I used in a class with some success. The enrollees were middle school and high school teachers, who were in their first year of teaching (not having arrived with necessarily strong coursework in mathematics, but let us not diverge for the moment).
Based on the interests of the (Boston-based) class, I phrased it in terms of scalping (Red Sox) baseball tickets. Students seemed engaged, and one of them emailed me the next day to say:
For what it's worth: I felt like Will Hunting when I got that Red Sox ticket markup problem yesterday.
(So, some anecdotal evidence that it was received okay!)
I phrase it below in terms of expensive jeans at a vintage clothing store, but I'm sure you can identify the underlying mathematics and modify the context as you see fit.
Real Life A: Your boss at a vintage clothing store tells you a particular brand of jeans have become very popular, and to mark them up by 10%. A few weeks pass, and she asks you to mark them back down by 10%. Is the final price more than, less than, or the same as the original price?
Real Life B: Same as A but the scenario is marking down by 10% first, and back up by 10% later.
Real Life C: Suppose the jean price in A is initially $200; leaving aside whether this is an exorbitant amount to pay for a pair of pants, what will be the final price?
It is a pretty common misconception that the before- and after-prices will be the same, so you could weave this into a tale of your boss (or maybe a nefarious manager, or whomever else) chastising you for messing up the prices. ("I told you to mark it back up by 10%; they should all say $200!" etc.)
Part C is probably the easiest among the three, since it deals with concrete numbers. But I have a habit of wondering about "inverting" problems; in the $200 case of C: Can you use the percentage increase and decrease to determine the final price? With an eye towards inversion, this led me to ask whether you could use the final price to determine the percentage increase and decrease:
Real Life D: A friend of yours is explaining that she was asked to mark a pair of $200 jeans up by some percentage, then back down by the same percentage, and her boss is convinced something went wrong because the final price is now less than the original. The boss is even talking about docking her pay, which seems unfair since your friend has done nothing wrong; she followed directions exactly, and now the jeans are priced 98 cents less than they were at the outset.
In scenario D, this is where you'd tune out and wonder:
Just what was the percentage by which the jeans were marked up and then marked down?
This is the kind of problem that (I think) might fascinate a mathematically inclined individual (especially if there was scrap paper -- or a napkin -- nearby). I think it is a nice place to point out where one might become curious and ask a mathematical question; even if your students do not share in this mathematical curiosity initially, it is a habit of mind that I think is worth fostering. Plus, you can try to work it out and then check correctness by asking your friend for the percentage at the end!
Omitting units, and an explanation around the setup, here is one algebraic approach:
$$200(1+p)(1-p) = 199.02 \iff 200(1-p^2) = 199.02 \iff 200 - 200p^2 = 199.02$$
$$\iff 0.98 = 200p^2 \iff 0.0049 = p^2 \iff 0.07 = p$$
So, the percentage change is 7%.
The initial equation is a true quadratic that can be solved with the quadratic equation or a calculator or whatever else; the solving method above proceeds by drawing from the knowledge around the product $(1+p)(1-p)=(1-p^2)$, and there is a lot more than be unpacked from the given approach: figuring out how to write the initial equation, figuring out how to solve it by hand (how do you hand compute $0.98/200$? well, you could divide by $2$ then divide by $100$), possibly thinking of $0.0049 = 49/1000$ for which finding the sqrt to be $7/100 = 0.07$ might be easier, investigating the other possible "solution" of -7%, and, more generally, this problem has all the components that I think one hopes a student can do by the end of an algebra unit on solving quadratics:
Solve a problem in more than one way (e.g., the method above as well as with the quadratic equation, though a thoughtful version of guess-and-check by recalling C and trying another value like 5% can work, too); represent a real life problem using mathematical notation; and sense-making fluently with decimals, fractions, and percentages in carrying out the algebra.