### Video Transcript

In this video, we will learn how to
compare and order rational numbers in different forms to solve real-world
problems. If this box is for all numbers, we
can divide them into two categories: rational and irrational. Rational numbers can be written as
a fraction, and irrational numbers cannot be written as a fraction. Integers would be rational
values. For example, negative three is an
integer. If we were going to write negative
three as a fraction, we could write it as negative three over one, which makes it a
rational number. Also, decimal values that terminate
are rational. 0.25 in fraction form is
one-fourth.

Irrational numbers cannot be
written as a fraction. An example of this would be đťś‹ or
the square root of two. We wonâ€™t be dealing with any
irrational numbers here. Weâ€™ve just said that rational
numbers are values that can be expressed as a fraction. So maybe we should remind ourselves
what is a fraction.

A fraction compares a part to a
whole. In a fraction, the denominator is
the value below the fraction bar, and it tells us the number of equal shares in the
whole. The numerator is the value on top
of the fraction bar, and it tells us the number of shares we are considering.

We commonly represent fractions
with a diagram using a shape, either a circle or a rectangle. The number of pieces the shape is
divided into represents the denominator. Here, our rectangle is divided into
five even pieces, making our denominator five. The piece weâ€™re considering is
usually shaded. Since two out of the five pieces
are shaded, we would say that this fraction is two-fifths.

Hereâ€™s an example of a circular
fraction diagram. The denominator is four, which
represents the number of equal shares in the circle. And the numerator would be three
because three of the pieces are shaded. This is the fraction
three-fourths.

But what if this diagram looked
like this, favorite color, and then had the labels blue and yellow? Just like before, the blue is
three-fourths, which means that the yellow is one-fourth. Itâ€™s very obvious here that
three-fourths is more than one-fourth. What this means is that ordering
fractions with the same denominator is very easy.

To compare five-eighths,
one-eighth, and three-eighths, the way we would order these values is to look at the
numerator. Five is the largest numerator. Therefore, out of these three
fractions, five-eighths is the largest. Three is the next largest of the
numerators, which makes three-eighths the second-largest fraction out of this
set.

But we need to consider what about
when the denominators are not the same. Comparing fractions with different
denominators would be a bit like this.

Daniel has nine coins, and Will has
three notes. If someone said, â€śNine is greater
than three; therefore, Daniel has more money,â€ť theyâ€™ve made an error in their
comparison because of the units. You canâ€™t compare the number of
coins to the number of notes. You would either need to convert
Danielâ€™s coins into the number of notes he would have or convert the amount of money
Will has into coins.

When we deal with fractions that
donâ€™t have the same denominator, weâ€™ll have to do the same thing. Weâ€™ll have to convert them to a
value that we can compare. Letâ€™s consider an example where we
would need to do that.

Compare one-half to
two-eighths.

These two values do not have a
common denominator, which means that in order to compare them, weâ€™ll need to find a
common denominator. Both of these values are even
numbers. In fact, two is a factor of
eight. If we multiply two by four, we get
eight. But itâ€™s very important when weâ€™re
working with fractions if we multiply our denominator by a value, we multiply the
numerator by that same value. One times four is four. This tells us that four-eighths is
an equivalent fraction to one-half.

Since now the two values do have a
common denominator, we look to their numerators. Four is greater than two. Four-eighths is larger than
two-eighths, which tells us that one-half is greater than two-eighths. This was the method of finding a
common denominator.

But we couldâ€™ve solved this another
way. We could have converted these
fractions to decimals. One-half written as a decimal is
0.5. And I see that the numerator and
the denominator of two-eighths are divisible by two. So we can say that two-eighths
equals one-fourth, and one-fourth written as a decimal is 0.25. If we compare the two decimal
values, weâ€™ll see that 0.5 is greater than 0.25, since 0.5 has a five in the tenths
place and 0.25 has a two in the tenths place. This confirms what weâ€™ve already
said, that one-half is greater than two-eighths.

Now, letâ€™s move on to an example
where weâ€™re comparing more than two values.

Arrange one twelfth, one-tenth,
one-third, and one twentieth in ascending order.

First, we need to know what
ascending order means. That would be from the least to the
greatest. And then we notice that we do not
have a common denominator. But all four values have a
numerator of one. Now, of course, we could find a
common denominator or convert all values to decimals. But because all four of the values
have the same numerator, we can arrange these four values with a different
method.

Imagine weâ€™re going to represent
these fractions with a circle diagram. The circle cut into thirds would
look like this. Here is that same circle cut into
12 pieces. The larger denominator has smaller
portions. One-third is much larger than one
twelfth. And so we can say when the
numerators are the same, the fraction that has the largest denominator will be the
first one when youâ€™re putting it from least to greatest. In this case, we would start with
one twentieth as the smallest value. And then we would move up to one
twelfth followed by one-tenth and then one-third. Weâ€™re able to do this kind of
comparison because the numerators were the same.

You might want to think about it as
the more pieces you cut the whole pizza into, the smaller the pieces get. So if you ate one slice of pizza
but there were only three slices, that would be more than if you cut the same pizza
into 10 slices and only ate one of the slices. For this question, in ascending
order, we get one twentieth, one twelfth, one-tenth, and then one-third.

Hereâ€™s another example of ordering
rational numbers. This time, none of the values are
given in fraction form.

Arrange the following in ascending
order: 0.2, negative 0.2, negative 2.3, nine, two.

We know that ascending order means
least to greatest. It might be helpful here for us to
think of these values in terms of a number line. We know all the positive values
will go to the right of zero and all the negative values will go to the left of
zero. When weâ€™re dealing with values that
are negative, values to the left of zero, the larger they are, the further away from
zero they will be. So negative 2.3 will be further
away from zero than negative 0.2. So we would call negative 2.3 the
least value.

After that would come negative
0.2. From here, we need to compare our
positive values. The decimal value is the only value
thatâ€™s less than one. So it would come next. And we know that two is smaller
than nine. So we would write two and then
nine. In ascending order, these values
are negative 2.3, negative 0.2, 0.2, two, and nine.

Hereâ€™s an example that has mixed
numbers and decimal values.

Arrange the following set of
numbers in ascending order: negative three and three-tenths, negative 3.61, and
negative 3.5.

Ascending order is least to
greatest, and we have three values. All three of the values are
negative. Two of them are decimal numbers,
and one is a mixed number. We have two choices. We could convert the mixed number
to a decimal, or we could convert both of the decimal numbers into mixed
numbers. Since two of the values are already
in decimal form, letâ€™s go ahead and convert this mixed number to a decimal
number.

We know that the negative three
will remain the whole-number portion. But how do we convert three-tenths
to a decimal? Since this is a fraction out of 10,
thatâ€™s a really simple procedure. We put a three in the tenths place
so that we have negative 3.3. At this point, we have to be really
careful because weâ€™re ordering negative values. And when we order negative values,
the least value will be the one thatâ€™s the furthest away from zero. And we know that the distance from
zero is the absolute value of our number.

That means negative 3.61 is 3.61
units away from zero to the left. And it means that negative 3.5 is
3.5 units away from zero to the left. Negative 3.5 is closer to zero than
negative 3.61, and negative 3.3 is the closest to zero of the three values. And that means to arrange them from
least to greatest, we would list negative 3.61, negative 3.5, and then negative
3.3. However, for the final answer, itâ€™s
best to take negative 3.3 and put it back in the format we were given: negative
three and three-tenths.

Hereâ€™s an example with both
positive and negative values and with mixed numbers and fractions.

Arrange the elements in the set one
and two-thirds, negative one-eighth, one and one-ninth, and negative one-half in
descending order.

Descending order is from greatest
to least. And so we notice about all of these
values that they do not have a common denominator. But if we think about them in terms
of the number line, we know that negative one-half needs to go to the left of zero
and one and two-thirds needs to go to the right of zero. At this point, weâ€™ll need to
compare negative one-half and negative one-eighth and one and one-ninth and one and
two-thirds.

Starting with the negatives, we
have negative one-half and negative one-eighth. If we multiply the numerator and
the denominator by four, for negative one-half, it becomes negative four over
eight. Now, because these values are
negative, we need to be very careful how we compare them. Negative one-eighth will be closer
to zero on a number line than negative four-eighths. So they belong on the number line
like this. If weâ€™re going in descending order,
from greatest to least, negative one-eighth would belong on the list before negative
one-half.

But now we need to compare the two
positive values. One and one-ninth and one and
two-thirds both have the whole-number portion of one. And that means to compare them,
weâ€™ll simply need to compare their fractions. These two fractions do not have a
common denominator. But if we multiply two-thirds by
three in the numerator and the denominator, this mixed number becomes one and
six-ninths. One and six-ninths is larger than
one and one-ninth. And so we could place them on our
number line like this. And weâ€™re ready to write them in
descending order.

Descending order is greatest to
least. We want to again write it in the
set notation, so weâ€™ll open the brackets. The largest of the values is one
and two-thirds, then one and one-ninth, then negative one-eighth, and negative
one-half. Weâ€™ll close the brackets. And weâ€™ve rearranged this set into
descending order.

Before we finish the video, letâ€™s
consider the key points we need to compare and order rational numbers. Rational numbers are real numbers
that can be expressed as simple fractions, meaning the denominator and the numerator
are integers. To compare fractions, one method is
to rename the fractions with common denominators. Another method is to convert all
values to decimals. Now youâ€™re ready to try some on
your own.