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
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
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
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