Just multiply both sides by 15.
You do still need to simplify by cancelling the 3, which technically is a division, so I’d say you’re saving one step rather than two.
How about the concept of averages, especially the arithmetic mean and geometric mean of a set of numbers? I would say that is pretty important and useful.
I would say both knowing averages, and knowing how to do them easily. If I have the numbers 171, 172, 173, 174, and 175, and I want the average, I’m not going to add them all together and divide by 5.
Geometric mean, though, I’d say isn’t nearly as generally applicable as arithmetic mean. What’s more useful is understanding arithmetic mean as compared to median.
Thanks. I really thought I was missing something.
I use it (or a variant) when adding up reps in crossfit. Where the workout is, for example, 3 reps, then 6 reps, then 9, reps for a certain number of rounds, but the score is total number of reps completed. I just do (first reps + last reps) * rounds/2 in my head, and the other people in the class think I’m some kind of wizard.
A wizard like Gauss?
You want to know how bad? I once saw a secretary take out out a calculator to divide something by 10. But the most incredible thing I ever witnessed happened when a secretary asked a grad student who happened to be in the math office what was 75 x 8 and she took out her calculator to do it. A grad student in mathematics. Later I asked my wife (who is not a mathematician) to do the problem and she answered instantly that it was 3/4 of 800 and therefore 600.
For services, I tip (nominally) 20% if the service was good. So say the restaurant bill is $63:
- Calculate 10% of $63 ($6.30).
- Double it ($12.60).
- Round up ($13).
I’ll use that as a starting point, at least. And then may make adjustments.
Why don’t you just divide by five?
Because calculating 10% takes no thought: just move the decimal point once to the left. And doubling it is also super easy.
YMMV, of course.
I’m decent at math, but I have poor visualization of problems like that. I’d have to write it on paper to look at it or do it with a calculator.
I just asked the PhD sitting next to me. She got it in 3 seconds, She said she doubled 75, then added 150 2 more times in her head.
I’m the same way; hearing numbers and trying to manipulate then just doesn’t work for me. I have two STEM degrees, I did a lot of math in my life…and I pull out calculators for simple stuff that my 10 year old can do without thinking.
I can do the math on paper, just not in my head. I never could. As a kid, I’d look at the ceiling and try to mentally write out problems and my father would tease me that the answers weren’t up there!
The algorithms you learned to use to multiply numbers on paper (at least, if you learned the same ones I did back in elementary school) don’t work all that well for performing mental calculations. So, if you’re trying to multiply 75 x 8 by visualizing “8 x 5 is 40; put down the 0 and carry the 4…” it’s understandably difficult. Mental calculation is almost a different skill than pencil-and-paper arithmetic.
To multiply 75 x 8, I’d employ one of the following methods:
- 75 = 100 – 25, so multiply 8 x 100 and subtract 8 x 25: 800 – 200 = 600.
- 8 = 2 x 2 x 2, so double 75 to get 150, double it again to get 300, and double it again to get 600.
@Hari_Seldon described how his wife did it, which is yet a third way.
I’ve seen discussions where people disparage “common core math” for having kids do calculations in weird, complicated ways that differ from what the older generation learned, but I think the point is to develop number sense and get them to be able to do calculations in ways like these that, depending on the circumstances, can be easier.
Some methods come to mind easily with some problems, and other ones take longer. Congrats on doing yours so fast.
Yes indeed. These methods wouldn’t necessarily have worked with different numbers. And having methods “come to mind easily,” and mental calculation in general, is a skill that gets easier with practice.
I don’t disagree with you and I can follow the logic really well in those examples, but I’d probably need a cheat sheet to help walk me through how to do it that way.
I suppose I could give myself homework and practice this skill, but it’s not going to happen.
I see the way my son is being taught at school and it does seem so much better. It makes sense, but old habits die hard. I have my phone and calculator app, and often my 10 year old who demands math questions as a hobby. I’ve gotten this far, I’ll be ok!
while on the topic … for a 112 year old, those numbers might just make their day …
so, the same math. concept might be good news or bad news, depending on where you stand …
yep, esp. when people (knowingly or ignorantly) confuse a variable’s manifestation (e.g. speed) with its 1st derivative (change of speed / acceleration or deceleration)…
so, no Mr. Politician … Inflation falling from 10% to 5% is not really good news and does especially NOT mean that bread will be cheaper in the future … It just means the price of bread will increase less than it did in the period before.
In one of his memoirs, Feynman wrote of being challenged to a mental-math competition, and winning handily, and saying that he was lucky that his opponent picked numbers for which there were easy tricks. But the thing is, Feynman knew so many tricks that, for nearly any problem one might pick, he probably would have known some trick or other for it.
I can do the mental math, but I’m lost once I get to first year Algebra.