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

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

There are two ways of describing the difference between integers: π‘Žβˆ’π‘. We can think of this as adding the additive inverse: π‘Ž+(βˆ’π‘), or we can think of this as the difference between their displacements from 0 on a number line.

Since we move from 0 to π‘Ž and then remove the displacement from 0 to 𝑏, we can think of this as the displacement from 𝑏 to π‘Ž.

We can use both of these ideas to consider the subtraction of rational numbers. If π‘Žπ‘ and 𝑐𝑑 are rational numbers, then we define their difference to be adding the additive inverse: π‘Žπ‘βˆ’π‘π‘‘=π‘Žπ‘+ο€»βˆ’π‘π‘‘ο‡.

Since the negative of a rational number switches the direction of its displacement on a number line, this is equivalent to finding the difference between the displacements of π‘Žπ‘ and 𝑐𝑑.

This time, we can note that π‘Žπ‘βˆ’π‘π‘‘ is the displacement from 𝑐𝑑 to π‘Žπ‘.

This gives us two ways of illustrating the difference of two rational numbers on a number line. Let us look for instance at the subtraction 12βˆ’13. We will first do this by considering this expression as 12+ο€Όβˆ’13. We then recall that we can add rational numbers if they have the same denominator. Since the lowest common multiple of 2 and 3 is 6, we can rewrite both rational numbers to have a denominator of 6: 12+ο€Όβˆ’13=1Γ—32Γ—3+ο€Όβˆ’1Γ—23Γ—2=36+ο€Όβˆ’26.

We then note that βˆ’26=βˆ’26, and we can add the numerators to get 36+ο€Όβˆ’26=36+βˆ’26=3+(βˆ’2)6=16.

This difference between 12 and 13, 16, can be illustrated as either what is added to 13 to reach 12 or the number that is reached from 12 going by 13 in the negative direction.

We see that we can add 16 to 13 to get to 12.

We can also note that 12βˆ’13 is the displacement from 13 to 12.

We want to find the rational number with an equivalent displacement from 0 so that we can evaluate this difference.

To do this, we can note that when sketching 13 on the number line, we split each integer segment into 3 sections with equal widths, and when sketching 12 on the number line, we split each integer segment into 2 sections with equal widths. If we instead split these into 6 sections, we can note that both of these points will lie on the increments and that these increments are of width 16.

Finally, since the displacement from 13 to 12 is 1 increment of 16, we can conclude that 12βˆ’13=16.

It is worth noting that this method takes more work than using a purely numerical approach. However, this does give a useful justification of why the numerical approach works.

We can simplify this process by noting that π‘Žπ‘βˆ’π‘π‘=π‘Žβˆ’π‘π‘.

We can then use this process to find the difference between rational numbers in general.

Definition: Difference between Rational Numbers

For any rational numbers π‘Žπ‘ and 𝑐𝑏, we have π‘Žπ‘βˆ’π‘π‘=π‘Žβˆ’π‘π‘.

For any rational numbers π‘Žπ‘ and 𝑐𝑑, we have π‘Žπ‘βˆ’π‘π‘‘=π‘Žπ‘‘βˆ’π‘π‘π‘π‘‘.

It is also worth noting that we can also use the lowest common multiple of 𝑏 and 𝑑 as the common denominator when subtracting fractions. For example, we can use the above definition to evaluate 12βˆ’14 as follows: 12βˆ’14=1(4)βˆ’2(1)2(4)=4βˆ’28=28.

We then cancel the shared factor of 2 in the numerator and denominator to get 12βˆ’14=14.

However, we can simplify this process by noting that the lowest common multiple of 2 and 4 is 4. Thus, we rewrite 12 to have a denominator of 4: 12βˆ’14=1Γ—22Γ—2βˆ’14=24βˆ’14.

Now, we evaluate the subtraction since they have a common denominator: 24βˆ’14=2βˆ’14=14.

This is the most general way to determine the difference between rational numbers.

In our first example, we will evaluate the difference between two rational numbers given as fractions.

Example 1: Subtraction of Rational Numbers as Proper Fractions with Like Denominators

Evaluate 25βˆ’45 giving the answer in its simplest form.

Answer

We first note that both rational numbers have the same denominator. We can then recall that if π‘Žπ‘ and 𝑐𝑑 are rational numbers, then π‘Žπ‘βˆ’π‘π‘=π‘Žβˆ’π‘π‘.

Hence, 25βˆ’45=2βˆ’45=βˆ’25=βˆ’25.

In our next example, we will see how we can evaluate the difference of rational numbers given as mixed fractions.

Example 2: Subtraction of Rational Numbers as Mixed Numbers with Unlike Denominators

Calculate 714βˆ’458. Give your answer as a mixed number.

Answer

We could start by converting both mixed numbers into fractions; however, this is not necessary. Instead, we first note that we can evaluate the subtraction of the whole parts and fractional parts separately since 714βˆ’458=714+ο€Όβˆ’458=ο€Ό7+14+ο€Όβˆ’4βˆ’58=(7βˆ’4)+ο€Ό14βˆ’58.

We have that 7βˆ’4=3, so we need to determine 14βˆ’58. To do this, we need to rewrite both fractions to have the same denominator. We can do this by noting that 4 is a divisor of 8, so the lowest common multiple of the denominators is 8. Thus, 14βˆ’58=1Γ—24Γ—2βˆ’58=28βˆ’58=2βˆ’58=βˆ’38.

Substituting in these values gives (7βˆ’4)+ο€Ό14βˆ’58=3+ο€Όβˆ’38.

We could now evaluate this expression by writing both numbers as fractions. However, we want to give our answer as a mixed number. So instead, we split 3 into 2+1 as shown: 3+ο€Όβˆ’38=2+1+ο€Όβˆ’38=2+ο€Ό1βˆ’38.

Now, we can rewrite 1 as 88 and evaluate to get 2+ο€Ό1βˆ’38=2+ο€Ό88βˆ’38=2+8βˆ’38=2+58=258.

In our next example, we will find the difference between rational numbers where one is given as a decimal.

Example 3: Finding the Difference between Rational Numbers Given in Different Forms

Evaluate 0.65βˆ’15, giving the answer as a fraction in its simplest form.

Answer

We could answer this question in different ways. For example, we could convert both rational numbers into decimals and then evaluate the subtraction. However, since we need to give our answer as a fraction, we will instead convert into fractions.

We first note that 0.05=5100=5Γ—15Γ—20=120.

We can then use the fact that 5 goes into 65 thirteen times to rewrite this as a fraction. We have 0.65=13Γ—0.05=1320.

Thus, 0.65βˆ’15=1320βˆ’15.

To subtract these fractions, we want to write them with a common denominator. We can do this by noting that 5 is a factor of 20. Therefore, 15=1Γ—45Γ—4=420. We can substitute this into the expression and evaluate to get 1320βˆ’15=1320βˆ’420=13βˆ’420=920.

It is also worth noting that we could answer this question by converting 15 into the decimal value of 0.2, since 15=20100=0.2, and calculating that 0.65βˆ’15=0.65βˆ’0.2=0.45=920.

In our next example, we will determine the difference between two rational numbers, one of which is given as a percentage.

Example 4: Finding the Difference between Two Rational Numbers by Converting Both into Decimals

Consider that π‘₯=34 and 𝑦=30%. By converting both π‘₯ and 𝑦 into decimal form, find the value of π‘₯βˆ’π‘¦ in decimal form.

Answer

We are told to evaluate this expression by first converting both numbers into decimals. We first note that 34=3Γ—254Γ—25=75100=0.75. We then recall that we can convert a percentage into a decimal by dividing by 100; we have 30%=30100=0.3.

Thus, π‘₯βˆ’π‘¦=0.75βˆ’0.3=0.45.

In our next example, we will evaluate an expression involving the difference of four rational numbers.

Example 5: Evaluating the Difference of Four Proper Fractions

Evaluate 13βˆ’512+38βˆ’16, giving the answer as a fraction in the simplest form.

Answer

We first note that we can evaluate this expression in any order by using the properties of the addition of rational numbers since subtracting a rational number is the same as adding its additive inverse. In particular, this means we can evaluate the sum or difference of multiple fractions by writing them to have the same denominator and then evaluating the numerators.

We can combine all of these fractions by rewriting them to have a common denominator. We see that 3 and 6 are both factors of 12, so we need to find the lowest common multiple of 8 and 12. We can calculate that this is 24. Thus, we can rewrite all fractions to have a denominator of 24 as follows: 13βˆ’512+38βˆ’16=1Γ—83Γ—8βˆ’5Γ—212Γ—2+3Γ—38Γ—3βˆ’1Γ—46Γ—4=824βˆ’1024+924βˆ’424.

We then rewrite this as a single fraction and evaluate as follows: 824βˆ’1024+924βˆ’424=8βˆ’10+9βˆ’424=324.

Finally, we cancel the shared factor of 3 in the numerator and denominator: 324=18.

In our final example, we will evaluate an expression involving the difference of three rational numbers given in different forms.

Example 6: Evaluating the Difference of Three Rational Numbers Given in Different Forms

  1. Express 5% as a fraction in its simplest form.
  2. Using the fraction form of 5%, find 5%βˆ’320βˆ’0.35. Express your answer as a fraction in its simplest form.

Answer

Part 1

We first recall that we can write a percentage as a fraction by dividing the numerical value by 100; this gives 5100. We can simplify this fraction by noting that 5 is a factor of 5 and 100. Hence, 5100=1Γ—520Γ—5=120.

Part 2

Since we need to evaluate this expression as a fraction, we will first write all of the terms as fractions. We already found that 5% is 120; we can also show that 0.35=35100=720. Thus, 5%βˆ’320βˆ’0.35=120βˆ’320βˆ’720.

Now, since the denominators are all equal, we can combine this difference into a single fraction by evaluating the numerator as follows: 120βˆ’320βˆ’720=1βˆ’3βˆ’720=βˆ’920.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • We can subtract rational numbers on a number line by finding the difference in their displacements from 0.
  • The difference between any two rational numbers always gives a rational number.
  • We can subtract rational numbers with the same denominator by subtracting their numerators: π‘Žπ‘βˆ’π‘π‘=π‘Žβˆ’π‘π‘, where π‘Ž, 𝑏, and π‘βˆˆβ„€ and 𝑏≠0.
  • To subtract rational numbers with different denominators, we can rewrite both fractions to have the same denominator by finding the lowest common multiple of the denominators. In general, for any rational numbers π‘Žπ‘ and 𝑐𝑑, we have π‘Žπ‘βˆ’π‘π‘‘=π‘Žπ‘‘βˆ’π‘π‘π‘π‘‘.
  • We can find the difference between rational numbers in different forms by converting both rational numbers into the same form.

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