### Video Transcript

In this video, we will learn how to
subtract rational numbers, including fractions and decimals. Before we consider subtraction of
rational numbers, let’s remind ourselves of what rational numbers are.

A formal definition of rational
numbers says that rational numbers are numbers that can be written in the form 𝑝
over 𝑞 where 𝑝 and 𝑞 are integers and 𝑞 is not equal to zero. Rational numbers can be made by
dividing two integers 𝑝 over 𝑞. Another way to think of that is
they can be written as a fraction of two integers. A value does not have to be in this
form to be rational. For example, 0.5 is not currently
written in the form 𝑝 over 𝑞, but we know that five-tenths is equal to
one-half. And because 0.5 can be written in
the form one over two, it’s rational.

All of the integers, the set of
whole numbers and their opposites, are rational numbers. A subset of integers, which are
rational, are all the whole numbers. And a subset of the whole numbers
are natural numbers, sometimes called counting numbers. The natural numbers are the set of
numbers starting with one and continuing up by one. Whole numbers include the value
zero. Integers include negative
values. And rational values can be decimals
as long as they can be written as fractions. For example, we know that 0.3
repeating is equal to one-third. Therefore, 0.3 repeating is
rational.

Mixed numbers are rational and so
are improper fractions. But in this video, we want to talk
about subtracting rational numbers, which reminds me of something my grandma used to
say. Apples and oranges! Let me explain. This is just an expression for when
two groups or items cannot be practically compared. And when we’re subtracting rational
numbers, we don’t wanna try and subtract values that are not written in the same
form. For example, if we wanted to say
two and a half minus 0.3 repeating, we’re starting with a mixed number and trying to
subtract a recurring decimal. And that is apples and oranges.

However, if we wrote 0.3 recurring
as a fraction and we rewrote two and one-half as an improper fraction, they would
then be found in the same form, and we could subtract. So, the key to subtracting rational
numbers is first making sure you put your values in the same format. We have to subtract apples from
apples or oranges from oranges. So let’s look at some examples.

Evaluate two-fifths minus
four-fifths giving the number in its simplest form.

When we look at this expression
two-fifths minus four-fifths, both of our values are fractions. And they have a common
denominator. Since both fractions are in the
same form and have a common denominator, to subtract, we simply subtract their
numerators and the denominator stays the same. This means we’ll have two minus
four over five. We know that two minus four equals
negative two. And then, the denominator is
five. And so, we say two-fifths minus
four-fifths is equal to negative two-fifths.

In our first example, we were
subtracting two fractions. In our next example, we’ll think
about how we would go about subtracting a mixed number from a mixed number.

Calculate nine and three twelfths
minus two and eight twelfths. Give your answer as a mixed
number.

We’re trying to subtract a mixed
number from a mixed number. And in order to do that, there are
a few things we should think about. First of all, for the fraction
portions of our mixed number, do they have a common denominator? In this case, they do; both have a
denominator of 12. After that, we could convert both
of these mixed numbers into improper fractions. However, since our final answer
needs to be written in the form of a mixed number, there’s also another strategy we
could use.

We could try and subtract eight
twelfths from three twelfths, but we’ve run into a problem because eight twelfths is
larger than three twelfths. So, we’ll need to rewrite at least
the nine and three twelfths. One way to do that is to take one
away from our whole number nine and make that eight. And when we do that, we can add 12
to our numerator. This is because we’ve taken one
from our whole number, which is equal to twelve twelfths. And twelve twelfths plus three
twelfths equal fifteen twelfths. Eight and fifteen twelfths is equal
to nine and three twelfths. It’s just written in a different
format.

Then, we can say eight and fifteen
twelfths minus two and eight twelfths. In this case, we can subtract the
whole numbers and the mixed number pieces. To subtract eight twelfths from
fifteen twelfths, we say 15 minus eight equals seven. And the denominator stays the same,
12. And now, we work on the whole
number pieces. Eight minus two equals six. And that means nine and three
twelfths minus two and eight twelfths will equal six and seven twelfths. This is already in the form of a
mixed number. And seven twelfths cannot be
reduced any further, making six and seven twelfths our final answer.

In our next example, again we’ll
have two mixed numbers, but this time we’re not beginning with common
denominators.

Calculate seven and one-fourth
minus four and five-eighths. Give your answer as a mixed
number.

When we look at the expression
seven and one-fourth minus four and five-eighths, they’re both mixed numbers. However, the fraction portion of
the mixed numbers do not have a common denominator. And we know in order to work with
these values, we need them to be in the same format. Since we’re dealing with
denominators four and eight and we know that four is a factor of eight, four times
two equals eight, which means we could rewrite one-fourth as two-eighths. Once we do that, we have seven and
two-eighths minus four and five-eighths.

However, we’re still not quite
ready to subtract because we’re trying to take five-eighths from two-eighths. And that means that for our first
mixed number seven and two-eighths, we need to borrow from the whole number
portion. If we take one whole away from
seven, we leave six. And that whole value that we took
away is equal to the fraction eight-eighths. If we add two-eighths plus
eight-eighths, we get ten-eighths. The mixed number six and
ten-eighths is the same value as seven and one-fourth written in a different
format.

Now, we’re ready to subtract, six
and ten-eighths minus four and five-eighths. To subtract the fraction portion,
since we have common denominators, we’ll say 10 minus five. That is, we’re subtracting the
values of the numerator. And the denominator isn’t
changing. Ten-eighths minus five-eighths
equals five-eighths. And then, we subtract the whole
number portions. Six minus four equals two, giving
us the result of two and five-eighths. Five-eighths can’t be simplified
any further. Two and five-eighths is a mixed
number and is our final answer.

So far, we’ve only been calculating
values that are already given in the same format. In our next example, we’ll have to
find the difference between a decimal and a fraction.

Find the difference between
negative 0.85 and two-fifths giving your answer as a fraction in its simplest
form.

The difference between negative
0.85 and two-fifths can be written as negative 0.85 minus two-fifths. But once we get to this point, we
have a problem. And that is that our two values are
given in different formats, which means we have a choice to make. We can convert two-fifths to a
decimal so that we’re subtracting a decimal from a decimal. Or we can convert negative 0.85
into a fraction so that we’re subtracting a fraction from a fraction. Both methods will work, but we’ve
been told to give the answer as a fraction in its simplest form. And that means it’s probably worth
it to subtract a fraction from a fraction.

And that means we need to think
about how we would write negative 0.85 as a fraction. Since the five in negative 0.85 is
in the hundredths place, we say that this is negative eighty-five hundredths. And as a fraction that is negative
85 over 100. But before we move on, we might
want to see if we can simplify eighty-five hundredths. Both 85 and 100 are divisible by
five. Negative 85 divided by five equals
negative 17, and 100 divided by five equals 20, which means we’re trying to say
negative 17 over 20 minus two-fifths.

However, we now see that we don’t
have common denominators in our two fractions. But we know that five times four
equals 20. And that means we can multiply
two-fifths by four over four. Two times four is eight, and five
times four is 20. We now have negative seventeen
twentieths minus eight twentieths. Since we have common denominators,
we can do subtraction by subtracting the numerators, which will be negative 17 minus
eight. The denominator doesn’t change. That remains 20. Negative 17 minus eight will be
equal to the negative value of 17 plus eight. That’s negative 25.

So, we have negative 25 over
20. Both of these values are divisible
by five. Negative 25 divided by five equals
negative five. 20 divided by five equals four. And this means negative 0.85 minus
two-fifths is equal to negative five-fourths.

Now, we’re ready to look at another
example.

Evaluate seven-fourths minus
negative one-half, giving the answer in its simplest form.

We have the expression seven over
four minus negative one over two. We recognize that both of these
values are fractions. However, they do not have the same
denominator. And we know in order to add or
subtract fractions, we need to first find a common denominator. We’re looking for the least common
multiple between two and four, sometimes called the LCM. If we think about multiples of two,
we have two, four, and six. But four is already a multiple of
two. And that means the least common
multiple of two and four will be four.

We won’t make any changes to
seven-fourths, but we’ll rewrite negative one-half as a fraction with the
denominator of four. Two times two is four. And negative one times two is
negative two. This means our new expression will
be seven-fourths minus negative two-fourths. We also remember that when we’re
subtracting a negative value, we can rewrite that as addition. So, seven-fourths minus negative
two-fourths is equal to seven-fourths plus two-fourths.

Once we have fractions with a
common denominator, we add them together by adding their numerators. Seven plus two is nine. And the denominator doesn’t
change. Seven- fourths plus two-fourths is
equal to nine-fourths. We wanna give this answer in
simplest form. And that means we want to check and
see if there are any common factors in the numerator and the denominator. Nine and four don’t share any
common factors apart from one. And that makes the fraction
nine-fourths in its simplest form.

In our final question, we’ll put
all these skills together as we subtract values in three different formats.

Evaluate five twelfths minus
negative one-third minus 0.75.

In this expression, we have
fractions with uncommon denominators and we have a decimal value. In order to do this subtraction,
we’ll need all three of these values in the same format. And that means we’ll need to
rewrite each of these values as fractions with common denominators. Before we deal with common
denominators, let’s start with our decimal value and write it as a fraction.

0.75 has a five in the hundredths
place, which means, as a fraction, 0.75 can be written as 75 over 100. But both 75 and 100 are divisible
by 25. 75 divided by 25 equals three, and
100 divided by 25 equals four. Seventy-five hundredths equals
three-fourths. It’s also possible that this is a
value you already know. It’s a common decimal and a common
fraction. Seventy-five hundredths equals
three-fourths.

Our new expression is then
five-twelfths minus negative one-third minus three-fourths. And we need a common denominator
from 12, three, and four. To find this common denominator, we
want the least common multiple. If we list the multiples of three,
we have three, six, nine, 12, 15. We do the same thing for four,
where we have four, eight, 12, and 16. At this point, we recognize that 12
is a multiple of both three and four. And that means 12 will be the least
common multiple between these three values.

We then want to rewrite all of
these fractions with the denominator of 12. Five twelfths remains the same. At this point, we can also say that
subtracting negative one-third is the same thing as adding. If we want to write one-third as a
fraction with the denominator of 12, we need to multiply it by four over four. And then, to rewrite the fraction
three-fourths with the denominator of 12, we multiply the numerator and the
denominator by three, which means our new expression will be five twelfths plus four
twelfths minus nine twelfths.

Once we have common denominators,
we can add and subtract by just adding and subtracting the numerators. And the denominator wouldn’t
change. So, we have five plus four minus
nine over 12. Five plus four is nine, and nine
minus nine is zero. Zero twelfths equals zero. And that means five twelfths minus
negative one-third minus 0.75 equals zero.

Before we finish, let’s quickly
review the key points. When subtracting rational numbers
as fractions, including mixed numbers, write all values with a common denominator
using the least common multiple. When subtracting rational numbers
as decimals and fractions, write all values as decimals or, alternatively, write all
values as fractions with a common denominator.