For example, to do 682 × 387. Please, I do not want to know neither the result nor the mechanism, only which one is the best for mental calculators.
These are two different things and I do both at the same time.
Personally, I calculate from left to right. Always.
If I do a multiplication I usually do the cross multiplication.
Sometimes I use different/other ways as you can see in my old posts.
The title of this thread “Left to Right“ or “Cross-Multiplication“. Which one is better? is a strange one.
There is no either/or. I do both at the same time. The one is not better or worse conpared to the other.
In school most people learn to calculate from right to left.
However; in real life things are different. Usually the first 2-3 digits are the most important and the rest not so much.
An example. Let’s say I calculate a mortgage amount for a house I intend to buy.
I usually estimate my answer. Assume my estimate came to $1.000.
Calculating from left to right, I might have the first 2 digits as ‘98’ and my answer is 98*.**
I don’t care whether the mortgage amount is $980 or $989, so I can stop after the second digit of my answer.
Also; for the people who do right to left, try doing a square root or 2 from right to left.
Long division; also needs to be done from left to right.
However; If you want to do multiplication from right to left, nobody would stop you and it is not wrong.
When I says
Cross-Multiplication I means this:
and when I says
Left to Right I means purely left-to-right, without crossed-multiplication, like this:
@Kinma Thanks for the reply, though.
I think @Kinma was referring to still doing criss cross but LTR instead of RTL. What you’re showing in your second picture is a whole different technique and could also be done RTL as well. Basically the way they teach it in high school.
As far as approximating the answer here for 348x461 take either a third of 461 for roughly 150 because 348 is almost 333 or take half of 348 because 461 is almost 500 for roughly 175. Since the latter overestimates and the former underestimates just average the two for something a little more than 160 and add the three 0s at the end.
If you don’t need the exact 160,428 you’ll be pretty close by just taking a third and a half and averaging. I’d have said roughly 160,000 without really calculating.
I’d suggest criss cross LTR if you speak the answer and RTL if you can write the answer. RTL just handles the carry nicer for me personally, but I also understand what @Kinma is saying.
This is true. It does handle the carry better.
@benjamin1990, I don’t like the method that you call the ltr method.
It does result in the right answer, but it is just too much work!
Doing this mentally means you need to remember all digits. No shortcuts like i explained in previous posts.
Also I then need to work in a - for me - annoying order. Difficult to explain, but look at your second picture.
First step is working on the ten thousands, then thousands, then hundreds, then thousands, then hundreds, then tens, then hundreds, then tens, etc.
The cross method fixes this.
So now that your question is clear, my preference is the cross method.
What do you like better?
Well, the trouble is remember the digits you are multipliyng -and I’m trying to fix it through memory techniques- , not the multiplication itself, as far as I am concerned.
The “cross-multiplication” technique add the carries in very short working-memory which distract me, so I guess I would like to say I prefer “left-to-right”.
Please elaborate. I don’t quite understand what you mean.
Why is that distracting?
Let’s do a quick cross multiplication with carry:
First Cross RTL:
9x9 = 81. Write 1. Remember the carry 8.
9x9 + 9x9=162. add carry (8) = 170. Write 0. Carry = 17.
9x9 = 81. Add carry (17) makes 97.
Answer = 9801.
Since we are writing down the digits from RTL, we only need to remember the carry.
Now LTR (there is 2 ways of doing this . I will first take the one that adds a zero at each step):
9x9 = 81. Answer so far 81. Add zero: 810.
9x9 + 9x9=162. Add to 810 makes 972. Add zero makes 9720. No carry.
9x9 = 81. Add 81 to 9720 makes 9801. Carry from adding 80 to 20.
Answer = 9801.
Now LTR (second way of doing; use the zeros in the calculation):
90x90 = 8100. Answer so far 8100.
90x9 + 9x90=1620. Add to 8100 makes 9720.
9x9 = 81. Add 81 to 9720 makes 9801. Only here we get a carry when adding the tens.
Answer = 9801.
Now your LRT method:
90 X 90 = 8100
90 X 9 = 810
Add gives 8910
9 X 90 = 810
Add to 9810 gives 9720. Carry from the hundreds into the thousands.
9x9 = 81
Add: 9801. Carry from the tens.
What distract me is the fact that I sometimes mix the carries with the multiplicand or the multiplier.
Is all about memory, not the operation of multiplication by itself.
Of course “LTR” isn’t better than “cross-multiplication” but since I have tried it so long, then I prefer it.
For a 2x2 I form the cross product first. Then the two outer products and only then do I sum.
Why? Because it is the most work?
Kinma, there is a Book about HIGH SPEED MATHEMATICS written by Trachtenberg. It will blow your mind. I wish you good luck !!!
Can you share a link to wikipedia or somewhere explaining. For me cross multiplication was only with fractions.
Yeah, it’s not called that, but @benjamin1990 posted an image of what he meant by it above. It’s usually referred to as criss-cross. You can find posts about it here on the forum as well…
I read it. It did not blow my mind.
Don’t get me wrong; I liked it.
I even adapted it for people who calculate from left to right:
and made an example based on the adaptation:
Just how is this relevant in this thread?