One-step multiplication and division equations
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One-step multiplication equations: fractional coefficients
- 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.