### Video Transcript

If 𝑎 is equal to a half 𝑏 and 𝑐
is equal to 𝑎 plus 𝑏, then express the ratio 𝑎 to 𝑏 to 𝑐 in its simplest
form.

A ratio is said to be in its
simplest form when all of our values are integers with no common factor apart from
one. In this question, we need to write
the ratio in the form 𝑎 to 𝑏 to 𝑐. We know that 𝑎 is equal to a half
of 𝑏. If we multiply both sides of this
equation by two, we get two 𝑎 is equal to 𝑏 or 𝑏 is equal to two 𝑎. We will call this equation one.

We are also told that 𝑐 is equal
to 𝑎 plus 𝑏. As 𝑏 is equal to two 𝑎, we can
substitute this into the equation. This means that 𝑐 is equal to 𝑎
plus two 𝑎. 𝑐 is therefore equal to three
𝑎. We will call this equation two.

As previously mentioned, we need to
write the ratio in the form 𝑎 to 𝑏 to 𝑐. We know that 𝑏 is equal to two 𝑎
from equation one. From equation two, we know that 𝑐
is equal to three 𝑎. The ratio can therefore be
rewritten as 𝑎 to two 𝑎 to three 𝑎. All of our terms have a common
variable 𝑎. This means we can divide them all
by 𝑎.

The ratio of 𝑎 to 𝑏 to 𝑐 is
therefore equal to one to two to three. As these three values have no
common factor apart from one, the ratio is in its simplest form.