### Video Transcript

π΄, π΅, πΆ, and π· are four points
on a straight line in that order. π΄π΅ to π΅π· is five to three. π΄πΆ to πΆπ· is three to one. Find the ratio π΄π΅ to π΅πΆ to
πΆπ·.

Now, the first thing Iβve done is
just drawn a representation of actually what this could look like, so our line π΄,
π΅, πΆ, and π·. Itβs not to scale. But it gives us an idea of where
they might be. First of all, if we think about our
ratio π΄π΅ to π΅π·, then this is five to three. So Iβve marked on our diagram, so
the five for π΄π΅ and then π΅ to π· is three. And then, we also have our ratio
π΄πΆ to πΆπ· and this is three to one. And again, Iβve actually marked
this onto our diagram.

Well, the first thing we know and
we can actually summarize from our diagram is that π΄π΅ is going to be five-eighths
of the total length and Iβve marked that on there. This is gonna be useful and Iβll
show you why now. Well, also, in addition, we know
that πΆπ· is a quarter of the total length. So now, weβve actually got two
parts and we know the fraction of the total of length that they are.

So therefore, using that, what we
can actually do is actually work out the fraction that π΅πΆ is going to be because
π΅πΆ is gonna have to be one minus five over eight minus one-quarter. So therefore, we can actually
calculate that because itβs gonna be one minus five-eighths minus two-eighths and
thatβs because a quarter is the same as two-eighths. And what Iβve done is multiplied
the numerator and denominator by two. So therefore, we can say that π΅πΆ
is gonna be one-eighth of the total length. And thatβs because weβve got π΅πΆ,
at one-eighth, plus π΄π΅, at five-eighths, makes six-eighths plus πΆπ·, at a
quarter, which is two-eighths makes eight-eighths or one whole. So that is the total length.

So now, we take a look back at the
question and we see whatβs it actually looking for. What itβs looking for is the ratio
π΄π΅ to π΅πΆ to πΆπ·. Well, if we think about it again,
if we got it in this order, weβve got π΄π΅ is five-eighths to π΅πΆ which is
one-eighth to πΆπ· which is two-eighths.

Therefore, we get the final answer
which is our ratio π΄π΅ to π΅πΆ to πΆπ· which is five to one to two. And weβve actually got this ratio
because we had five-eighths to one-eighth to two-eighths and they all have the same
denominator. So we can simplify that to five to
one to two.