One-step multiplication equations: fractional coefficients | 6th grade | Khan Academy

Let’s say that we have the equation two-fifths x is equal to ten. 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.

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 ten; two times five is ten. So it's going to be ten over ten, or one.

You could think about five divided by five is one, two divided by two is one. 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. 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, 1x is equal to 10 times five-halves. That's the same thing as fifty-halves. I could write it this way, fifty over two, which is the same thing as twenty-five.

Let's do another example. Let’s say we have the equation fourteen is equal to seven-thirds b. See if you can solve this. Well, once again it would be nice if the coefficient on the b weren't seven-thirds, but instead we're just one. If you just said 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 the 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 a b, and fourteen is three-sevenths times fourteen.

You might see this as fourteen over one, and you could say, okay, this is going to be three times fourteen over seven times one. Or you could say hey, let's divide both a numerator and a denominator by seven. So this could be two, and this could be one. So you're 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 six. 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.

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.