I got my first calculator in jr. high in the early-’70s. It had four functions: Addition, Subtraction, Multiplication, and Division. It had an 8-digit nixie tube display, used (IIRC) eight AA cells, and cost $99. I recall it got rather warm, and burned through AA cells pretty quickly.
What do you mean by “know their times tables”? And is that a question about “pencil-and-paper arithmetic” or about “mental math” or about remembering basic facts?
I think that it’s more important to be able to see that both fractions are a bit less than 1/2 and that they are fairly close to each other in value, and to know how to tell which is larger with a calculator available.
Please concentrate more on the trivial calculations. 
The answer is 9¢, not .1¢.
Good catch. It is indeed nine cents, not one tenth of a cent!
The math that the cashier would need to do is still taught. I would not even assume she couldn’t do that math ordinarily. She easily could have had trouble thinking of it due to the script being flipped on her. She’s not used to having to think about the transaction, as the machine tells her what to give you back, and she’d now have to think about “how do I handle changing the amount he gave me?”
I suspect in other words it could be less practical knowledge of need to use it, and just the rut of working a job, possibly on low sleep. And not that she couldn’t handle it in a math classroom.
Good point.
“sig figs” as my college prof put it.
The two errors cancel out, don’t they?
Know the product of any pair of single digits, instantly – no thought involved. Pencil-and-paper long division takes five? ten? times as long, if you have to figure out each multiplication involved.
By the way – my question isn’t about what kids should know. All I’m asking is: what do they know, nowadays?
When I’m teaching Algebra II and Precalculus students about polynomials, and specifically division of polynomials, I always start by asking them, first, how many of them learned long division of numbers (almost all hands go up), and second, how many of them actually remember it (maybe a third of the hands go up). And of that second group, it’s probably a smaller number yet that actually remember it.
Anyway, I re-teach it anyway, because most students definitely don’t remember it (the technique for dividing polynomials is almost identical to long division of numbers).
I’ve got a math minor and a EE degree, but even back then we were using calculators for most of our long division problems.
I still can do long division, but it’s not something I’m super fast with.
“New Math”, by Tom Lehrer
Don’t judge math skills by a cashier. Fast food places all switched in the 1980’s to cash registers with item icons, automatically calculated totals and maybe even change (?). It was a marked change from the 70’s style where one had to punch in each individual item price.
Abacus skills seem to generally no longer taught to school kids throughout Asia these days. Japan still seems to teach: “Soroban remains part of the national curriculum in many elementary schools.” (Copilot)
There’s a famous proof that most adults cannot understand fractions.
A hamburger chain tried to beat McDonalds by selling a 1/3 pounder at the same price as McD’s quarter pounder. People refused to buy it, because they thought 1/3 was less than 1/4.
Trial & Error is not far from the “official” method.
Make a guess. Divide the number by the guess. Average the 2 numbers. Repeat until the 2 numbers are “close enough.”
Okay, show me. Here’s how I did it. Each line is equivalent to the line above.
16/37 < 29/67 iff
37/16 > 67/29 iff (subtract 2 from each side)
5/16 > 9/29 iff
16/5 < 29/9 iff (sub 3)
1/5 < 2/9 iff
5 > 9/2 which is obviously true.
But I don’t think I could carry this out in my head.
I cross-multiplied (see post 11).
But I like the continued fraction method, too.
I did the same basic thing as Chronos without the fancy square roots and mental arithmetic. It may be worth spelling it out for anyone getting lost.
29/67 ? 16/37
(29*37)/(67*37) ? (16*67)/(37*67) make the denominators the same
29*37 ? 16*67 ignore the denominators now that they match
1073 > 1072 do the two multiplications (on paper)I don’t recall ever learning this at school, so the whole discussion of that equation has had me mystified from the beginning. And I think of myself as reasonably academically smart.
“Long division” means long division:
11.09053
----------
123.00
1
---
023
21
---
200
0
---
20000
19881
-----
11900
0
-----
1190000
1109025
-------
8097500
6654309
If you look at the Harvard admissions exam from 1869, page 6, or the one from MIT, you can see that finding square roots and cube roots to a few places of decimals was on the test, though the method is not specified: it just asks to show your work.
