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### Course: Numbers & Operations - The Real & Complex Number Systems 211-217 > Unit 6

Lesson 1: Multiplication as scaling# Multiplication as scaling with fractions

Learn about the concept of multiplication as scaling. Watch and understand how multiplying fractions can be seen as scaling, or resizing, the value of a number. Visualize this concept with various examples, reinforcing their understanding of multiplication as a scaling process. Created by Sal Khan.

## Want to join the conversation?

- hi is this easy for anyone because sadly i DO NOT GET IT!(109 votes)
- Hi! I don't know if your question has been answered yet, but if not, maybe this will help. What Sal is trying to explain in the video is that anytime you multiply a number, let's say x, by another number, let's say m, that is less than 1, then the product will be less than the original number x. That is, given a positive integer x, and another positive integer m, and m<1, then mx < x.

Now if you multiply a number, x, by a number, m, that is greater than 1, then their product will be greater than x. That is, if x is a positive integer, and m is a positive integer and m>1, then mx > x.

Finally, if you multiply a number x by a number m that is equivalent to 1, then the product of the numbers will be equivalent to x. That is, if x is a positive integer, and m=1, then mx = x.

SO..... to relate it to the video, in each scenario Sal gives, x = 2/3. So that is the positive integer that you will multiply by other positive integers to compare their products. But the point he is trying to make is that you don't actually have to multiply in order to know if the product will be less than, greater than, or equal to 2/3.

In the first example, 2/3 (remember, x) is being multiplied by 7/8. So the m I was talking about before is 7/8. Now 7/8 is less than 1, so 2/3*7/8 < 2/3 (if you want to prove it to yourself, 2/3*7/8 = 14/24 = 7/12, and so you can compare 7/12 with 2/3, 2/3=8/12, so we can now see that 7/12 is in fact less than 8/12).

In the second example, 2/3 is being multiplied by 8/7. So m=8/7 now. Since 8/7 > 1, then 2/3*8/7 > 2/3. Again, you can prove that to yourself by actually doing the math like I did in the paragraph above.

Finally, in the last example, even though it doesn't initially look like it, 2/3 is being multiplied by 5/5. Now 5/5 = 1, so 2/3*5/5 = 2/3.

Sorry for the lengthy explanation, but hopefully that's helpful to someone out there!(92 votes)

- I don't want to watch this but i have to i have no choice.(20 votes)
- Yup, The videos are kind of confusing, wish they changed that.(12 votes)

- I am stuck in this section and really worried, what is the exact way for knowing that which number is greater than 1 or less than

1/3 x 575, 4/3 x 575, 5/6 x 575. how can we know that which one of them greater or lesser (1/3, 4/3, 5/6)I am stuck in this section and really worried, what is the exact way for knowing that which number is greater than 1 or less than

1/3 x 575, 4/3 x 575, 5/6 x 575. how can we know that which one of them greater or lesser (1/3, 4/3, 5/6)I am stuck in this section and really worried, what is the exact way for knowing that which number is greater than 1 or less than

1/3 x 575, 4/3 x 575, 5/6 x 575. how can we know that which one of them greater or lesser (1/3, 4/3, 5/6)(18 votes)- Here's how to find if a fraction is greater than 1 or less than one. Let's look at your fractions. You have 1/3, 4/3, and 5/6. There are 2 numbers in each fraction. Let's look at the first one. 1/3 has 1 as the numerator, and 3 as the denominator. If the numerator is smaller than the denominator, it's less than 1. If the numerator is larger than the denominator, it's larger than 1. So 1/3 and 5/6 are both less than 1, and 4/3 is greater than 1. Now time for the comparison. 5/6 is the odd one out when we're looking at denominators. Let's look at the LCM (Least Common Multiple). 6 is the LCM, so let's convert the denominators with 3 to 6, by multiplying both the numerator and denominator by 2, so the new fractions are 2/6, 8/6, and 5/6. Now we just order all of the fractions by their numerator, so the least to greatest is 2/6, 5/6, and 8/6.

Sorry for the lengthy explanation, but it's the easiest way to give the answer, and to the point where something is learned!(2 votes)

- Wait is this basically just multiplying fractions?(10 votes)
- I don't understand this, What is scaling.(10 votes)
- A scale factor is a number which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x.(5 votes)

- but 3x5=15 so it is equal for 15/15 that's a whole(10 votes)
- It is a whole number because 7/7 well you divide 7 and 7 so you get 1 but if somewone asked what 1 x 1/1 that one will be 1/1 x 1/2 so it equals 1/2(4 votes)

- Whats 9 plus 10?(0 votes)
- If you break it down and start by crossing out 2/3 from each expression, you can see the
had the lowest value because its factor 7/8 was**first expression****less than 1**. Thehad the highest value because its factor 8/7 was**second expression****greater than 1**, and thewas in the middle because it was**third expression****equal to 1**, being greater than 7/8 and less than 8/7. So from least to greatest, it would be 7/8, 1, 8/7. Then don't forget to multiply 2/3 back to each number to get the full expression back.(5 votes)

- so, if you are multeplying by a number smaller than 1 you scale down? so the middle one would be the smallest since 2/3 is smaller than 1?(5 votes)
- why is it just times two thirds why not other numbers?(4 votes)
- In my opinion its easier to visualize and to comply with, unlike for beginners where if a beginner faces a problem like 68/89 it would be harder for them since its bigger.(4 votes)

## Video transcript

We have three expressions here. This is 2/3 times 7/8. The second expression
is 8/7 times 2/3. This third expression is
5 times 2 over 3 times 5. And what I want you to do is
pause this video right now and think about which
of these expressions is the largest, which one is in
the middle in terms of value, and which one is the smallest. And I want you to think
about it without actually doing the calculation. If you could just look
at them and figure out which of these is the largest,
which of these is the smallest, and which of these
is in the middle. So pause the video now. Now, you might have
taken a shot at it. And I'll give you a little
bit of a hint in case you had trouble with it. All of these involve
multiplying something by 2/3. And you see a 2/3 here. You see a 2/3 here. And it might not be as obvious,
but you also see a 2/3 here. And let me rewrite that to
make it a little bit clearer. So this first expression could
be rewritten as 7/8 times 2/3. This second expression here
could be written as-- well, it's already written
as 8/7 times 2/3. And then, this
last expression, we could write it as, in
the numerator, 5 times 2. And then in the denominator,
it's over 5 times 3. 5 times 3, which is of course
the same thing as 5/5 times 2/3. So you see, all three
of these expressions involve something times 2/3. Now, looking at
it this way, does it become easier to pick
out which of these are the largest, which of
these are the smallest, and which of these are
someplace in between? I encourage you
to pause it again if you haven't
thought about it yet. So let's visualize each
of these expressions by first trying
to visualize 2/3. So let's say the height of
what I am drawing right now, let's say the height of this
bar right over here is 2/3. So this right over
here represents 2/3. The height here is 2/3. So first, let's think about
what this one on the right here represents. This is 5/5 times 2/3. Well, what's 5/5? 5/5 is the same thing as 1. This is literally
just 1 times 2/3. This whole expression
is the same thing as 1 times 2/3, or
really, just 2/3. So this, the height
here, 2/3, this is the same thing as
this thing over here. This is going to
be equal to-- this could also be viewed as
5 times 2 over 3 times 5, which was this first
expression right over here. Now, let's think about
what these would look like. So this is 7/8 times 2/3. So it's less than 8/8 times 2/3. It's less than 1 times 2/3. So we're going to
scale 2/3 down. This is going to
be less than 2/3. It's going to be 7/8 of 2/3. So this one right over here
would look something like this. Let me see if I can draw it. Yeah, it would look
something like this. If the yellow height is 2/3,
then this right over here, then this height right over
here-- let me make it clear. This height right over here
would be 7/8 times 2/3. Likewise, let's look at
this one right over here. Let's look at this one in
the middle, 8/7 times 2/3. Well, 8/7 is bigger than 7/7. It's more than 1. This is more than 2/3. This is 1 and 1/7 times 2/3. So it's going to be the same
height as 2/3 plus another 1/7. So it's going to look
something like this. It's going to look
something like this. So its height-- now we
scaled the 2/3 up because 8/7 is greater than 1. So this right over
here, this height is going to be 8/7 times 2/3. So the way that you
could have spotted which of these is the
largest and which of these is the smallest is to say, well,
how are they scaling 2/3? This one right over
here, you're essentially multiplying 2/3 by 1. So you're just going to get 2/3. You're not scaling it up, or
you aren't scaling it down. This one right over here,
you're scaling 2/3 down. You are multiplying it
by something less than 1. If you multiply it by
something less than 1, then you're going to
be scaling it down. I should say, a positive
number or a number between 0 and 1--
less than 1-- then you're going to be
scaling it down. So this thing is scaled down. It's going to be the smallest. And here, you're multiplying
the 2/3 times a number bigger than 1, by 1 and 1/7. So you're going to scale it up. So this expression is the
largest, 8/7 times 2/3. The smallest is 2/3 times 7/8. And this one right over
here is in between.