Main content

## Pre-algebra

### Course: Pre-algebra > Unit 7

Lesson 4: One-step multiplication and division equations- One-step division equations
- One-step multiplication equations
- One-step multiplication & division equations
- One-step multiplication & division equations
- One-step multiplication & division equations: fractions & decimals
- One-step multiplication equations: fractional coefficients
- One-step multiplication & division equations: fractions & decimals

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# One-step multiplication equations: fractional coefficients

Learn how to solve equations with fractions by using reciprocals. Multiply both sides of the equation by the reciprocal of the fraction to make the coefficient one. This simplifies the equation, making it easier to find the value of the variable. Created by Sal Khan.

## Want to join the conversation?

- why r some vids so long(9 votes)
- Maybe because you need to understand what it is.(9 votes)

- Math can be confusing and this is very hard Help meeee. This also makes 0 sense.(7 votes)
- another tip when multiplying fractions by fractions if there is an expression is to:

multiply d by d and n by n.

for fraction by fraction division you can do the first n divided by the second d and the second n by the first d then multiply the 2 fractions you got.(4 votes)

- I tried calculating the first sum on my own and did a completely in a different way compared to the way Sal had explained and demonstrated it...And I checked if my answer was correct and it was so....What should I do? Should I keep doing it the way I did it or should I do it the way Sal had?(4 votes)
- My advice is that you should try both ways first, and then go with the answer that feels right to you, and then check your answer. Have a nice day!(5 votes)

- ngl this was a really helpful video(6 votes)
- How do you get 12/15 from 10.6= 12/15c/2?

I don't understand at all.

At5:13Sal says,

"To get a variable do it".

How does that make any sense?(5 votes) - Can someone else explain this to me shorter and easier to understand(2 votes)
- Basically, if you have a variable with a coefficient that is a fraction, such as 1/4x = 5, you can multiply both sides by the reciprocal of the coefficient to find what the variable is.

1/4 x 4/1 = 1

5 x 4/1= 20

x = 20

Hope this helped!(8 votes)

- Why is there no comments 🤨(1 vote)
- It is because this is a new video. Many people have not watched it yet.(2 votes)

- idk how this is makes any sense.??(2 votes)
- 4/3b = 4/5

What is b?

I’m very confused, I found the LCM of them, but i have to multiply 20/15 by some number to get 12/15. What? ? ?(1 vote)- Do you understand one step equations with division or multiplication? If you have 3x = 9, coefficients indicate multiplication, so opposite is division, divide by 3 on both sides, 3/3 x = 9/3. If you have x/4=2, opposite of divide is multiply, so multiplying by 4 gives 4^x/4=2*4, so x=8. If you understand these two, fractions are just combining these two separate things into 1, if you have 3/4 x = 15, then multiply by 4 and divide by 3 gives 4/3 * 3/4 x = 15* 4/3, so x=20. Opposites of fractions are their reciprocal.(2 votes)

## Video transcript

- Let's say that we have the equation, two-fifths X is equal to 10. How would you go about solving that? Well, you might be thinking to yourself, it would be nice if we just had an X on the left-hand side
instead of a two-fifths X, or if the coefficient on the X were one instead of a two-fifths. And the way that we might do that, is if we were to multiply both sides of this
equation by five halves. Why five halves? Well, five halves, if you notice, when I multiply five
halves times two-fifths, it's going to get us to one. Five times two is 10, two times five is 10. So it's going to be 10 over 10 or one, or you could think about
five divided by five is one, two divided by two is one. And you might say, "Is that magical? How did you think of five halves?" Well, five halves is just
the reciprocal of two-fifths. I just swapped the numerator
and the denominator to get five halves. And then why did I multiply
it times the right-hand side? Well, anything I do to the left hand, I also want to do to the right hand. So the left-hand side
simplifies to this is all one. So it's just going to be X is equal to, or we could say one X is
equal to 10 times five halves. That's the same thing as 50 halves. I could write it this way, 50 over two, which is the same thing as 25. Let's do another example. Let's say we have the equation, 14 is equal to seven-thirds B. See if you can solve this. Well, once again, it'd be nice if the coefficient on the B weren't seven-thirds, but instead were just a one. If it's just B is equal to something. Well, we know how to do that. We can multiply both
sides of this equation times the reciprocal
of the coefficient on B times the reciprocal of seven-thirds. What's the reciprocal of seven-thirds? Well, the denominator
will become the numerator. The numerator becomes a denominator. It's going to be three-sevenths. Now, of course, I can't just do it on one side. I have to do it on both sides. So on the right-hand
side of this equation, three divided by three is one, seven divided by seven is one. Those all cancel out to one. So you're just left with one B or just to B and 30, or three-sevenths times 14, you might see this as 14 over one. And you could say okay, this is going to be three
times 14 over seven times one, or you could say, hey, let's divide both a numerator
and denominator by seven. So this could be two.
And this could be one. So your left with three
times two over one times one which is just going to be equal to six. Let's do another example. Let's say that we had one sixth A is equal to two-thirds. How could we think about solving for A. Well, once again, it would be nice, if this one-sixth were to become a one and we could do that by multiplying by 6. Six-sixths is the same thing as one. And to make it clear that
this is the reciprocal, we could just write six wholes as six ones or six wholes when you multiply these, this is all going to be equal to one. So you're left with one
A on the left-hand side, but of course, you can't just
do it on the left-hand side. You have to also do it
on the right-hand side. So A is going to be equal to, over here we could say two
times six over three times one. So that would be twelve-thirds or we could say, look, six and three are
both divisible by three. So six divided by three is two, three divided by three is one, two times two is four over one times one. So it's going to be four wholes
or just four and we're done.