Question Video: Subtracting Mixed Numbers in a Real-Life Application

Video Transcript

Liam had a piece of lumber 15 feet long. He cut some pieces to use in his workshop and three and a half feet remained. How much of the wood did he use?

So originally, Liam had 15 feet of lumber. However, after he cuts some pieces to use for his workshop- so we took some of the wood away and we don’t know how much that is. So we could say minus 𝑥. And we know that what’s left is three and a half feet.

So we need to find how much wood he used, which is 𝑥. So we need to solve for 𝑥. And we can do so by bringing 𝑥 to the right-hand side of the equation because it’s negative on the left-hand side. So the 𝑥s cancel on the left. And now, to get 𝑥 all alone, we need to subtract three and a half feet from both sides of the equation.

However, how do we subtract three and a half from 15? Well, we could take 15 and then subtract three, which would be 12. However, we have another half that we need to take out. So 12 minus one-half is 11 and a half. But how we actually get that? Well, we can do this two ways. We could make 15 and three and a half both be improper fractions or we could have 15 be changed a little bit.

So let’s begin by making them both be improper fractions. 15 is the same as 15 over one. Now, three and a half, there’s a little trick. We take two times three which is six and then add the one which is seven. And we keep our denominator of two. So it’s the same as seven-halves. Or another way to think about that is we can make three have a denominator of two. Three is the same as six-halves. So if we have three and a half, we need to take six-halves and add one-half, which is seven-halves.

So to take 15 minus three, we could take 15 over one minus seven over two. However, they have to have a common denominator, which will be two. So we need to take 15 times two and one times two. So 30 halves minus seven halves means we’ll keep our common denominator of two and take 30 minus seven, which is 23.

Now, let’s change this back to a proper fraction. So two goes into 23 evenly 11 times. And then, one-half will be left over because two goes into 22 11 times. So just as we said before, the amount of wood that he would have used would be 11 and a half.

Now, the other way that we could have solved this would be to rewrite 15 to have a little fraction beside it. So we could have 15 be changed to be 14 and one over one because one over one is actually just one and 14 plus one is 15.

Now, the whole reason for doing this is that way we can subtract the one-half from something. So we have one over one as a fraction and we have one-half as a fraction. So to make them the same, let’s change one over one to have a denominator of two. So we would have to multiply the top and bottom by two.

And now, we’re ready to subtract. So 14 minus three — the whole numbers — is 11. And two over two minus one over two is where we subtract the numerators, so two minus one which is one. And we keep the denominator of two. So we get 11 and a half.

So once again, we have a final answer of 11 and a half feet. That’s how much wood that Liam used.