Fraction multiplication as scaling examples

This right over here is an image from an exercise on Khan Academy, and it says compare using greater than, less than, or equal to. On the left, we have one fourth times five thousand, and we want to compare that to five thousand.

On Khan Academy, you'd click on this drop down, and you would pick a greater than, less than, or equal to. So what do you think it is? How does one-fourth times five thousand compare to five thousand, and how do you know your answer is correct? Pick one of these choices as well. Pause the video and have a go at that.

Well, one way to think about it, if I'm multiplying five thousand times some value, that value is going to scale five thousand. It's going to make it smaller, or bigger, or in certain cases, might not change the size at all. In fact, the ones where you don't change five thousand is when you multiply it by one. So one times five thousand is equal to five thousand.

But what do you think is going to happen when you multiply a value that is between zero and one times five thousand? Is that going to be greater than, equal to, or less than five thousand? Well, one way to imagine it is on a number line. If this is zero, this is five thousand; this would be halfway between zero and five thousand.

If I then make the space between zero and five thousand into fourths, this would be one fourth of the way between zero and five thousand. So this value right over here, that is one fourth times five thousand. Well, how does this value, one fourth times five thousand, compare to this value, to five thousand?

Well, it's clear that one fourth times five thousand is less than five thousand. In general, if the value is between zero and one that you're multiplying by something else, it's going to make the thing you're multiplying smaller. If this were 1, it's not going to change the value, and if this were greater than 1, it would make it larger.

So let's see these choices. Because we are multiplying five thousand by a fraction that is less than 1, yeah, that's exactly what's happening. We're multiplying five thousand by one fourth, so it's going to give us a smaller value than five thousand. I like that choice.

Let's just read these other ones. Because we're multiplying five thousand by a fraction that is equal to one? Well, no, we clicked one fourth, which is clearly not equal to one, so we ruled that one out. Because we were multiplying five thousand by a fraction that is greater than 1? Well, one fourth is clearly not greater than 1, so we rule that out as well.

Let's do another example. So here we're asked if the product of each expression is less than, equal to, or greater than 49. So pause this video and try to figure it out.

All right, so this first one is 49 times seven-eighths. So what do you think that's going to be? Well, seven-eighths is less than 1. So if I multiply 49 times something less than 1, I'm going to get a value that is less than 49. So I like that choice right over there.

49 times five halves? Well, five halves is greater than 1. Remember, two halves would be equal to 1. So five halves is greater than 1. So if I'm multiplying 49 by a value that is greater than 1, I'm going to get a value that is greater than 49.

And then last but not least, 49 times three-thirds? Well, three-thirds is exactly equal to one, so 49 times 1 is going to be exactly equal to 49, and we're done.