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### Course: Arithmetic (all content) > Unit 5

Lesson 19: Multiplying whole numbers and fractions- Multiplying fractions and whole numbers visually
- Equivalent fraction and whole number multiplication problems
- Multiply fractions and whole numbers
- Multiply fractions and whole numbers with fraction models
- Equivalent whole number and fraction multiplication expressions

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# Multiplying fractions and whole numbers visually

Learn the concept of multiplying fractions and whole numbers. Watch how to visually represent this process and practice understanding the relationship between fractions and whole numbers in multiplication. Created by Sal Khan.

## Want to join the conversation?

- I'm just wanting to confirm and understand why:

2/5 = 2 x 1/5

and why isn't it like before with mixed numerals where this is done:

2 x 1/5 = ((2 x 5)+1)/5 = 11/5 ?

is there a conceptual difference i am missing? thanks in advance !! :)(73 votes)- Yes there's a conceptual difference.

I believe you are confusing multiplication of a whole number and a fraction, with addition of a whole number and a fraction. Many students have difficulty in math, including higher levels such as algebra, ultimately because they confuse the fundamental concepts of addition and multiplication.

((2 x 5)+1)/5 is equivalent to the mixed number 2 and 1/5. This mixed number means 2 + 1/5, not 2 x 1/5.

2 x 1/5 can be thought of as the repeated addition 1/5 + 1/5, which is clearly 2/5.(86 votes)

- *
*why can't we just add it the normal way.like 1/5+3/4=4/9**(19 votes)- You cant just add the denominators as well as adding the numerators, you need to find a common factor.(17 votes)

- If whatever happens to the denominator happens to the numerator why is the whole number only multiplied by the numerator and not the denominator? Are whole numbers +-x/ fractions only between the whole number and numerator?(8 votes)
- Technically, A whole number n is the same as n/1. Since anything multiplied by 1 is itself, The denominator doesn't change.

You only multiply both numbers by the whole number when you're finding a common denominator.(5 votes)

- If 1/2 x 5 is 5/2 does that mean 5 x 1/2 is the same?(5 votes)
- yes because 1/2 x 5 is the same as 1/2 x 5/1 (because 5 ones are five) and if you multipy 1 x 5 (the top) and 2 x 1( the bottom) you get 5/2 so you are correct(7 votes)

- I don't understand please help I am trying to get like what to do other than this fifths(6 votes)
- multiplying fractions are one of the hardest fractions but sometimes it can be easy for some people(3 votes)

- how would it be smaller instead of being bigger?(4 votes)
- this is because of the fact that the fraction is not equal to one, so it makes the number smaller. Think of these fractions as decimals, when you multiply a number by a decimal it becomes smaller. so 3 x 1/3 is 1, because it becomes 3/3 which is one(1 vote)

- why is it that when you add denominators together, they aren't different. For example: 2 fourths + 2 fourths do not equal 4 eighths. it equals 4 fourths instead. Why is that true?(3 votes)
- Try imagining that the denominator is telling us how big the parts are. Adding parts together doesn't the size of those parts individually, but changes their amount over-all.

Hope this helps.(3 votes)

- Does anyone have any questions about this lesson? I check my notifications almost every day, so if you have questions about this lesson, feel free to let me know in a reply to this comment!

PS: Put your question in an answer to this question so that I can reply to your question, I will still check the regular comments though. Also, if this was commented at least four months ago, your question will probably not be answered.(4 votes)- How did he get 6/5? I thought it was supposed to 6/30 because whatever you multiply to the top, you do the same to the bottom.(1 vote)

- before i watched the video i already knew that when u multiply a fraction u just get the whole number like this example:5 x3/4 the denominator stays the same and the numerator changes just have to multiply the whle number to the numerator which is 15/? then the denominator stays the same like this 15/4 so yea that what i knew before i watched the vid.then after i watched the vid i taught me the same thing but in a longer form.(4 votes)
- On the last fraction, why are all the one fifths in parentheses?(0 votes)

## Video transcript

We've already seen that the
fraction 2/5, or fractions like the fraction 2/5, can
be literally represented as 2 times 1/5, which
is the same thing, which is equal to literally
having two 1/5s. So 1/5 plus 1/5. And if we wanted
to visualize it, let me make a hole
here and divide it into five equal sections. And so this represents
two of those fifths. This is the first of the
fifths, and then this is the second of the fifths,
Literally 2/5, 2/5, 2/5. Now let's think about something
a little bit more interesting. What would 3 times
2/5 represent? 3 times 2/5. And I encourage you
to pause this video and, based on what
we just did here, think about what you think
this would be equivalent to. Well, we just saw that 2/5
would be the same thing as-- so let me just
rewrite this as instead of 3 times 2/5
written like this, let me write 2/5
like that-- so this is the same thing as
3 times 2 times 1/5. And multiplication, we can
multiply the 2 times the 1/5 first and then
multiply by the 3, or we can multiply the 3
times the 2 first and then multiply by the 1/5. So you could view this literally
as being equal to 3 times 2 is, of course, 6, so this is
the same thing as 6 times 1/5. And if we were to try
to visualize that again, so that's a whole. That's another whole. Each of those wholes
have been divided into five equal sections. And so we're going to
color in six of them. So that's the first 1/5, second
1/5, third 1/5, fourth 1/5, fifth 1/5-- and that
gets us to a whole-- and then we have
6/5 just like that. So literally 3 times 2/5
can be viewed as 6/5. And of course, 6
times 1/5, or 6/5, can be written as--
so this is equal to, literally-- let me do the
same color-- 6/5, 6 over 5. Now you might have said, well,
what if we, instead of viewing 2/5 as this, as we just
did in this example, we view 2/5 as 1/5 plus
1/5, what would happen then? Well, let's try it out. So 3 times 2/5-- I'll rewrite
it-- 3 times 2/5, 2 over 5, is the same thing as
3 times 1/5 plus 1/5. 2/5 is the same thing
as 1/5 plus 1/5. So 3 times 1/5 plus
1/5 which would be equal to-- well, I just
have to have literally three of these added together. So it's going to be
1/5 plus 1/5 plus 1/5 plus 1/5 plus-- I think
you get the idea here-- plus 1/5 plus 1/5. Well, what's this going to be? Well, we literally
have 6/5 here. We can ignore the parentheses
and just add all of these together. We, once again, have
1, 2, 3, 4, 5, 6/5. So once again, this
is equal to 6/5. So hopefully this
shows that when you multiply-- The 2/5 we saw
already represents two 1/5s. We already saw that,
or 2 times 1/5. And 3 times 2/5 is literally
the same thing as 3 times 2 times 1/5. In this case, that would be 6/5.