Hum
Went out for lunch today. It cost $8.23. I gave the cashier $10, then said "Oh, wait, I have the $.23," and gave it to her. She'd already rang the $10 in, and then had to call someone over to ask how much change to give me. I feel quite sorry for her - is she just confused by registers? innumerate? learning disability? or just not willing or able to go that one extra step in figuring something out?

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But this example... let's just hope she's a freshman when it comes to handling the register. Or very very bad at math. Or just very very dumb. :)
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Next you'll tell me you didn't do well in grammer school Latin classes, either.
Whatever happened to quality education?
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Worked one summer in a busy sandwich shop with an old-fashioned cash register that thought its job was done once it had calculated and added in the tax. Yes, I can do all these things involving change - despite the fact that I consider myself bad at math. (It's just calculation - it's not like, mathematics.) But I've seen a number of cashiers get confused by the kind of scenario you describe.
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In the early 90s I worked in an old Roy Rogers where all the cashiers had to calculate change themselves, even on the drive-through window where you'd have several orders waiting to be paid. The registers were that old and decrepit. As a result to this day I often give money to cashiers that would be easier for them to give me change from if only they understood math or money.
...god, I feel old.
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The "right" way is to say, "I've been given 10.23. The bill was 8.23. What do I give her back in return."
The way lots of people think about these things is, "OMG, the register says to give back 1.77, but she just added 23 cents to what she gave me!! WhatdoIdo?!?"
It probably doesn't help that manipulating the amounts given and received is a common scammer tactic.