Alternate mental subtraction method
Sal makes several arithmetic slips while talking: "898 minus 80 is 9" (he means 89 - 80), "15 minus 8 is 17" (he means 25 - 8), and "14 minus 8 is 16" (it is 6). The final answer 1,867 is correct. I added notes at those points.
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Sal shows a mental way to subtract: take away the bottom number one place at a time (thousands, then hundreds, tens, ones) from both numbers, so you only ever remember two numbers. He solves 9,456 - 7,589 = 1,867 this way and then checks it with the usual borrowing (regrouping) method.
You'll be able to
- subtract a large number in my head by taking it away one place value at a time.
- explain why this method only needs me to remember two numbers at each step.
- check a mental subtraction using the borrowing (regrouping) method.
Key words
- subtract
- To take one amount away from another to find the difference.
- borrowing (regrouping)
- Taking 1 from the next place to the left so a top digit becomes 10 more and you can subtract.
- place value
- The value a digit has because of its position, like thousands, hundreds, tens or ones.
Transcript
I want to show you a way that, at least, I find more useful to subtract numbers in my head. And I do it this way-- it's not necessarily faster on paper, but it allows you to remember what you're doing. Because if you start borrowing and stuff it becomes very hard to remember what's actually going on.
So let's try out a couple of problems. Let's have 9,456 minus 7,589. So the way I do this in my head. I say that 9,456 minus 7,589-- you have to remember the two numbers. So the first thing I do is I say, well, what's 9,456 minus is just 7,000? That's pretty easy because I just take 9,000 minus 7,000.
So what I can do is I'll cross out this and I'll subtract 7,000 from it. And I'm going to get 2,456. So in my head I tell myself that 9,456 minus 7,589 is the same thing as-- if I just subtract out the 7,000-- as 2,456 minus 589. I took the 7,000 out of the picture.
I essentially subtracted it from both of these numbers. Now, if I want to do 2,456 minus 589 what I do is I subtract 500 from both of these numbers. So if I subtract 500 from this bottom number, this 5 will go away. And if I subtract 500 from this top number, what happens?
What's 2,456 minus 500? Or an easier way to think about it? What's 24 minus 5? Well, that's 19. So it's going to be 1,956. Let me scroll up a little bit. So it's 1,956. So my original problem has now been reduced to 1,956 minus 89. Now I can subtract 80 from both that number and that number.
So if I subtract 80 from this bottom number the 8 disappears. 898 minus 80 is just 9. And I subtract 80 from this top number, I can just think of, well, what's 195 minus 8? Well, 195 minus 8, let's see. 15 minus 8 is 17. So 195 minus 8 is going to be 187 and then you still have the 6 there.
So essentially I said, 1,956 minus 80 is 1,876. And now my problem has been reduced to 1,876 minus 9. And then we can do that in our head. What's 76 minus 9? That's what? 67. So our final answer is 1,867. And as you can see this isn't necessarily faster than the way we've done it in other videos.
But the reason why I like it is that at any stage, I just have to remember two numbers. I have to remember my new top number and my new bottom number. My new bottom number is always just some of the leftover digits of the original bottom number. So that's how I like to do things in my head.
Now, just to make sure that we got the right answer and maybe to compare and contrast a little bit. Let's do it the traditional way. 9,456 minus 7,589. So the standard way of doing it, I like to do all my borrowing before I do any of my subtraction so that I can stay in my borrowing mode, or you can think of it as regrouping.
So I look at all of my numbers on top and see, are they all larger than the numbers on the bottom? And I start here at the right. 6 is definitely not larger than 9, so I have to borrow. So I'll borrow 10 or I'll borrow 1 from the 10's place, which ends up being 10.
So the 6 becomes a 16 and then the 5 becomes a 4. Then I go to the 10's place. 4 needs to be larger than 8, so let me borrow 1 from the 100's place. So then that 4 becomes a 14 or fourteen 10's because we're in the 10's place. And then this 4 becomes a 3. Now these two columns or places look good, but right here I have a 3, which is less than a 5.
Not cool, so I have to borrow again. That 3 becomes a 13 and then that 9 becomes an 8. And now I'm ready to subtract. So you get 16 minus 9 is 7. 14 minus 8 is 16. 13 minus 5 is 8. 8 minus 7 is 1. And lucky for us, we got the right answer. I want to make it very clear.
There's no better way to do this. This way is actually kind of longer and it takes up more space on your paper than this way was, but this for me, is very hard to remember. It's very hard for me to keep track of what I borrowed and what the other number is and et cetera.
But here, at any point in time, I just have to remember two numbers. And the two numbers get simpler every step that I go through this process. So this is why I think that this is a little bit easier in my head. But this might be, depending on the context, easier on paper.
But at least here you didn't have to borrow or regroup. Well, hopefully you find that a little bit useful.
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