Multiplying fractions and whole numbers visually
Sal gives a visual explanation of multiplying a fraction and a whole number. 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 !! :)(51 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.(57 votes)
- Sal explained it in a simple way, but it is confusing for me. I am here to know whether to do reciprocal in the multiplication of fractions. Can someone pls clarify my question?(20 votes)
- You do not have to use reciprocals when multiplying fractions, only when dividing.(6 votes)
- *why can't we just add it the normal way.like 1/5+3/4=4/9*(14 votes)
- You cant just add the denominators as well as adding the numerators, you need to find a common factor.(10 votes)
- How do you multiply by 1000000(2 votes)
- Any time you're multiplying by a number that is a 1 followed by some number of 0's, you can simply add on that amount of 0's. If I were to multiply 37 by 10, my answer would be 370. Multiplying 42 by 100 results in 4200. Multiplying 13 by 1 million (1000000) results in 13 million (13000000)(14 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?(5 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.(3 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 already know this for some reason(7 votes)
- Do they ever answer us back?cause i asked them something from like 3 years ago(6 votes)
- People DO answer your questions, but there are thousands of questions, so it's certainly possible that not every one gets an answer.
Many of the people answering these questions are those who have taken the course and have mastered all of the concepts - in other words former students who are volunteering their time to answer questions.(3 votes)
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.