Canceling zeros when dividing | Math | 4th grade | Khan Academy

Let's solve 350 divided by 50. So one way to think about this is if we had 350 of something, let's say something delicious like brownies. If we had 350 brownies and we were dividing them into groups of 50, how many groups of 50 could we get? Well, one idea is to count by 50s until we get to 350 and see how many groups there are. One group of 50 would be fifty, plus another group of fifty would be a hundred, plus another group of fifty is one hundred fifty. Another one will be two hundred, two hundred plus fifty is two hundred fifty, plus another 50 is 300, and plus one more 50 is 350.

So when we added all these 50s—these groups of 50: 50, 100, 150, 200, 250, 300, 350—we ended up with 350. So, 350 can be divided into this many groups of 50. And how many groups is this? 1, 2, 3, 4, 5, 6, 7. So, 350 divided into groups of 50 is 7.

Now let's look at that quotient, that solution of 7, and let's notice something. If we had simply divided 35 by 5, we would have also gotten 7. 35 divided into groups of 5 is also equal to 7. So in a way, these zeros didn't matter; they didn't affect our answer. We were able to cancel them out. Having the zero in the first number is cancelled out by having that zero in the second number.

And let's look at why that is. Let's think about why that is. Division is really the same as a fraction. So if we wrote this as a fraction—let's say 350 over 50—well, this fraction bar right here is the same as the division sign up here. 350 divided by 50 is the same as 350 over 50. And when we have a fraction like this, we can simplify it. In this case, because there are zeros on the end, we know they're both multiples of 10, so we can divide them both by 10.

We can divide our numerator and our denominator by 10. And when we divide whole numbers by 10, we have a trick we can use, a pattern really, which is that the whole number—in this case, 350—when it's divided by 10, we drop a zero from the end. So, 350 divided by 10 is 35. 350 could be divided into 10 groups of 35. And then 50 divided by 10 will be the same thing. When we divide 50 by 10, we drop that zero off the end. Or another way to think about it is 50 divided into groups of 10 would make five groups.

And then we end up with our simplified fraction of 35 fifths or 35 divided by 5, which is the same thing here. So, in both of these cases, we can see that 350 divided by 50 is the same as 35 divided by 5. So when both whole numbers—when we're dividing whole numbers and they both end in zeros—we can cancel those zeros. Basically, we're factoring out a 10. We're taking the 10, the divided by 10, out of both of them, out of both numbers. So we can cancel those zeros, which leaves us with smaller numbers. At least for me, I find division a lot simpler when I'm working with smaller numbers.

Let's try a few more. Let's try one like 420 divided by 70. So we can see we have two whole numbers, both end in zero, so we're going to cancel those zeros. Basically, we're dividing a 10 out of both numbers and end up with a simpler division problem of 42 divided by 7. 42 divided by 7 equals 6; therefore, 420 divided by 70 also equals 6.

And here's one last one. What if we had 5600 or 5600 divided by 80? So the first thing I notice is I have zeros at the end of both of my whole numbers. If I cancel out this one, I can cancel out one on the other side. I can't cancel both of them; over here, there were two. Canceling, we have to cancel the same amount of zeros on both sides, and now we end up with 560 divided by 8.

This leaves us with a new division problem that's still a little bit tricky but easier than dividing by 80. So here we can think of 560 as 56 tens because of the zero on the end. And 56 tens, or 560, could be rewritten as 10 times 56. These are equivalent: 560 and 10 times 56 because 10 times 56 is a 56 with a zero on the end. Rewriting it this way makes it so now I have a division problem right here: 56 divided by 8 that we can solve. 56 divided by 8 is 7, and then we still have this 10 and the times sign, and 10 times 7 equals 70.

So our solution here, when we divided 560 by 8, was 70. So that means our solution up here for 5600 divided by 80 is also 70 because when we cancel zeros and divide, we still get the same solution. But remember, we have to cancel the same amount of zeros in each number. Here, we couldn't cancel both zeros. If we cancel one zero in the dividend, we can cancel one zero in the divisor.

And another thing to remember about this trick: the first thing to remember is we need to cancel the same amount of zeros, and the second—the zeros have to be at the end of the problem. So if we had a division problem, something, for example, like 506 divided by 20, here we cannot cancel the zeros. We cannot cancel the zeros because this zero is not on the end. So the new problem this would give us, 56 divided by 2, is not equivalent, does not equal the top division problem. So we can't cancel zeros; that does not work. We cannot cancel zeros unless they are at the end of the problem, and we cancel the same amount in both the dividend and the divisor.