### Video Transcript

Find the rational number that lies
one-third of the way from the smaller number between negative one-half and negative
one and one-half.

In this question, we are asked to
find the rational number that lies between negative one-half and negative one and
one-half that lies one-third of the way from the smaller of these numbers to the
larger of these numbers.

To answer this question, we can
start by sketching both numbers on a number line to help us determine which is the
smaller number and how far is one-third of the distance. To do this, we first rewrite
negative one and one-half as negative three-halves. Since both numbers are multiples of
one-half, we can count in halves to the left of zero to represent negative one-half
and negative three-halves on the number line. We count to the left since both
numbers are negative. We obtain the following sketch. We can see that negative
three-halves is the smaller of the two numbers. And we can also note that the
distance on the number line between the two values is one, since we have to travel
two intervals of one-half to travel between them.

One-third of the distance between
the numbers is one over three, which is one-third. So we want to travel an interval of
one-third along the number line from negative three-halves. There are many ways of doing
this. Since we have a number line, we
will do this calculation on the number line. We can note that splitting halves
into thirds will give us sixths, since two times three is six. So, we will split the number line
into sixths. We can then note that adding
one-third is the same as adding two-sixths. So we need to move two increments
of one-sixth to the right of negative three over two. This gives us negative
seven-sixths.

Hence, negative seven-sixths lies
one-third of the way from negative one and one-half to negative one-half.