Lesson Video: Subtraction of Rational Numbers Mathematics • 7th Grade

In this video, we will learn how to subtract rational numbers, including fractions and decimals.


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.

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