𝐴, 𝐵, 𝐶, and 𝐷 are four points
on a straight line in that order. 𝐴𝐵 to 𝐵𝐷 is five to three. 𝐴𝐶 to 𝐶𝐷 is three to one. Find the ratio 𝐴𝐵 to 𝐵𝐶 to
Now, the first thing I’ve done is
just drawn a representation of actually what this could look like, so our line 𝐴,
𝐵, 𝐶, and 𝐷. It’s not to scale. But it gives us an idea of where
they might be. First of all, if we think about our
ratio 𝐴𝐵 to 𝐵𝐷, then this is five to three. So I’ve marked on our diagram, so
the five for 𝐴𝐵 and then 𝐵 to 𝐷 is three. And then, we also have our ratio
𝐴𝐶 to 𝐶𝐷 and this is three to one. And again, I’ve actually marked
this onto our diagram.
Well, the first thing we know and
we can actually summarize from our diagram is that 𝐴𝐵 is going to be five-eighths
of the total length and I’ve marked that on there. This is gonna be useful and I’ll
show you why now. Well, also, in addition, we know
that 𝐶𝐷 is a quarter of the total length. So now, we’ve 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 𝐵𝐶 is going to be because
𝐵𝐶 is gonna have to be one minus five over eight minus one-quarter. So therefore, we can actually
calculate that because it’s gonna be one minus five-eighths minus two-eighths and
that’s because a quarter is the same as two-eighths. And what I’ve done is multiplied
the numerator and denominator by two. So therefore, we can say that 𝐵𝐶
is gonna be one-eighth of the total length. And that’s because we’ve got 𝐵𝐶,
at one-eighth, plus 𝐴𝐵, at five-eighths, makes six-eighths plus 𝐶𝐷, 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 what’s it actually looking for. What it’s looking for is the ratio
𝐴𝐵 to 𝐵𝐶 to 𝐶𝐷. Well, if we think about it again,
if we got it in this order, we’ve got 𝐴𝐵 is five-eighths to 𝐵𝐶 which is
one-eighth to 𝐶𝐷 which is two-eighths.
Therefore, we get the final answer
which is our ratio 𝐴𝐵 to 𝐵𝐶 to 𝐶𝐷 which is five to one to two. And we’ve 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.