Subtracting Mixed Fractions

Mixed fractions are another form of representing an improper fraction composed of a whole number and a proper fraction. Subtracting mixed fractions is the subtraction operation performed between any two mixed fractions. We will be studying different methods and rules to understand subtracting mixed fractions in this article.

Two or more fractions having a common denominator are known as like fractions. Hence, mixed fractions with like denominators will have the same denominators such as \(3\dfrac{2}{7}\) and \(2\dfrac{1}{7}\). Look at the following points to be kept in mind while subtracting mixed fractions.

• A mixed fraction \(a\dfrac{b}{c}\) can also be written as a + (b/c).
• To convert a mixed number to an improper fraction, the whole number is multiplied by the denominator and the result is added to the numerator of the proper fraction by retaining the denominator. For example, to convert \(1\dfrac{6}{11}\) to an improper fraction, we multiply 1 and 11 i.e, 1 × 11 = 11 and the result is added to 6 i.e., 11 + 6 = 17. Thus the improper fraction is 17/11.
• To convert an improper fraction to a mixed number we will divide the numerator of the improper fraction by its denominator. The quotient becomes the whole number part, the remainder becomes the numerator of the proper fraction and the denominator remains as it is. For example, to convert 22/3 to a mixed number, we first divide 22 by 3 and get the quotient as 7 and the remainder as 1. Thus, the mixed number is \(7\dfrac{1}{3}\).

Now let us understand the steps of subtracting mixed fractions with like denominators.

Example: Subtract the mixed fraction \(2\dfrac{1}{3}\) from \(4\dfrac{2}{3}\).

We have to perform \(4\dfrac{2}{3}\) - \(2\dfrac{1}{3}\). Let's look into the steps.

• Step 1: We will subtract the whole numbers of both the fractions. i.e., 4 - 2 = 2.
• Step 2: Now we will subtract the fractional parts. i.e., (2/3) - (1/3) = 1/3.
• Step 3: We will combine the result of the last two steps to get the result. i.e., 2 + (1/3) = \(2\dfrac{1}{3}\).

Hence the value of \(4\dfrac{2}{3}\) - \(2\dfrac{1}{3}\) is \(2\dfrac{1}{3}\).

Fractions with unequal denominators are known as unlike fractions. Thus, some examples of mixed fractions with unlike denominators are \(5\dfrac{2}{3}\) and \(1\dfrac{2}{5}\). Let us take an example to understand the steps of subtracting mixed fractions with unlike denominators.

Example: Subtract \(3\dfrac{1}{6}\) from \(5\dfrac{2}{3}\).

We have to perform \(5\dfrac{2}{3}\) - \(3\dfrac{1}{6}\). We have two ways to perform the subtraction.

Method I: By subtracting the whole numbers separately and the fractions separately by making their denominators equal.

• Step 1: Subtract the whole numbers of both the fractions. i.e., 5 - 3 = 2.
• Step 2: Now we will subtract the fractional parts. To do so we have to make the denominators of 2/3 and 1/6 equal by finding their LCM.
• Step 3: Since the LCM of 3 and 6 is 6, we will write 2/3 as (2 × 2) / (3 × 2) = 4/6.
• Step 4: We will now subtract the fractions. i.e., (4/6) - (1/6) = 3/6.
• Step 5: The result obtained in the previous step will be simplified. i.e., 3/6 = 1/2.
• Step 6: The result of step 1 and step 5 will be combined to get the final result. i.e., 2 + (1/2) = \(2\dfrac{1}{2}\).

Method II: By converting them into improper fractions, followed by subtracting them by making their denominators equal.

• Step 1: Convert the given mixed fractions to improper fractions. i.e., \(5\dfrac{2}{3}\) = 17/3 and \(3\dfrac{1}{6}\) = 19/6.
• Step 2: For the obtained fractions in the last step, we will make the denominators equal by taking their LCM.
• Step 3: LCM of the denominators 3 and 6 is 6. Thus, 17/3 can be written as (17 × 2) / (3 × 2) = 34/6.
• Step 4: Now, we will subtract the fractions. i.e., (34/6) - (19/6) = 15/6.
• Step 5: The result of the previous step will be simplified. i.e., 15/6 = 5/2.
• Step 6: Finally we convert the result obtained in the last step to a mixed fraction. i.e., 5/2 = \(2\dfrac{1}{2}\).

Hence the value of \(5\dfrac{2}{3}\) - \(3\dfrac{1}{6}\) is equal to \(2\dfrac{1}{2}\).

While subtracting mixed fractions, there might arise a situation wherein the fraction to be subtracted is greater than the fraction from which it is being subtracted. In such cases, we will use the concept of regrouping. Let's now understand subtracting mixed fractions with regrouping by taking an example.

Example: Subtract \(7\dfrac{2}{3}\) from \(10\dfrac{4}{9}\).

We have to perform \(10\dfrac{4}{9}\) - \(7\dfrac{2}{3}\).

• Step 1: Consider the fractional parts of both the mixed fraction and compare them by making their denominators equal. i.e., we will be comparing 4/9 and 2/3.
• Step 2: The LCM of the denominators 9 and 3 is 9. Thus, 2/3 can be written as (2 × 3) / (3 × 3) = 6/9. Hence we see that 6/9 > 4/9 or we can say 2/3 > 4/9.
• Step 3: As seen in the previous step 4/9 < 6/9, we cannot subtract 6/9 from 4/9. Hence, now 4/9 will borrow 1 from the whole number part of the mixed fraction \(10\dfrac{4}{9}\).
• Step 4: The whole number 10 gives away 1 as borrow to 4/9. We know that 1 can also be written as 9/9. Hence, when the borrow 9/9 is added to 4/9, we get 4/9 + 9/9 = 13/9.
• Step 5: We will now rewrite the fraction after regrouping. The whole number 10 becomes 9 after giving a borrow to 4/9 and 4/9 becomes 13/9. Hence, \(10\dfrac{4}{9}\) = \(9\dfrac{13}{9}\).
• Step 6: Now we will subtract the mixed fractions easily as they have like denominators. i.e., \(9\dfrac{13}{9}\) - \(7\dfrac{6}{9}\) = \(2\dfrac{7}{9}\).

Related Articles on Subtracting Mixed Fractions

Check these articles related to the concept of subtracting mixed fractions.

• Mixed fractions
• Improper fractions
• Proper fraction
• Fractions

How to Solve Subtracting Mixed Fractions?

Subtracting mixed fractions can be done in two ways. For like denominators, the whole numbers can simply be subtracted and the fractional part of the mixed fractions can also be subtracted and the two results are combined to get the result. The other way to do it is, by converting the mixed fractions to improper fractions and subtracting them. For unlike denominators, they can be first converted to like denominators by finding the LCM and the same steps can be followed as subtracting mixed fractions with like denominators.

How to Borrow when Subtracting Mixed Fractions?

While subtracting mixed fractions, if the proper fractional part of the mixed fraction from which the other mixed fraction is getting subtracted is smaller, then the whole number gives a borrow to the proper fraction to make it larger. For example, to perform \(3\dfrac{1}{3}\) - \(1\dfrac{2}{3}\) we see that 2/3 > 1/3. Thus, 1/3 will borrow 1 whole from 3. 1 whole can be written as 3/3. The whole number 3 after giving a borrow of 1 becomes 3 - 1 = 2 and the fraction 1/3 becomes (1/3) + (3/3) = 4/3. Thus, the new modified mixed fraction after borrowing is \(2\dfrac{4}{3}\). Now, the subtraction will be \(2\dfrac{4}{3}\) - \(1\dfrac{2}{3}\) = \(1\dfrac{2}{3}\).

How to Regroup when Subtracting Mixed Fractions?

Regrouping is done when a greater fraction is subtracted from a smaller fraction. For example, let's perform \(8\dfrac{4}{9}\) - \(5\dfrac{2}{3}\) . We will be making the denominators of 4/9 and 2/3 equal to compare them. The fraction 2/3 can also be written as 6/9. But 6/9 > 4/9. We cannot subtract a larger fraction from a smaller fraction. Thus, 4/9 has to be made larger. To do so, 4/9 borrows a 1 from 8. 1 whole can also be written as 9/9. Now, the whole number 8 becomes 8 - 1 = 7 and the fraction 4/9 becomes (4/9) + (9/9) = 13/9. Thus, the new fraction will be \(7\dfrac{13}{9}\). Thus, now the subtraction is as follows: \(7\dfrac{13}{9}\) - \(5\dfrac{6}{9}\) = \(2\dfrac{7}{9}\).

How to Subtract Mixed Fractions with Same Denominators?

Subtracting mixed fractions with the same denominators is done by subtracting the whole number part and fractional part of the mixed fractions separately followed by combining them to get the result.
For example, let's perform \(23\dfrac{3}{4}\) - \(21\dfrac{1}{4}\)
= (23 - 21) + (3/4) - (1/4)
= 2 + (2/4)
= \(2\dfrac{2}{4}\)

On simplifying we get,
= \(2\dfrac{1}{2}\)

How to Subtract Mixed Fractions with Different Denominators?

Subtracting mixed fractions with different denominators can be done by converting them into an improper fraction followed by converting them into like denominators by taking their LCM and finally subtracting their numerators. The final result is then converted back to a mixed fraction.
For example, let's perform \(6\dfrac{2}{3}\) - \(2\dfrac{1}{4}\)
= (20/3) - (9/4)
= [(20 × 4) / (3 × 4)] - [(9 × 3) / (4 × 3)]
= (80/12) - (27/12)
= 53/12
= \(4\dfrac{5}{12}\)

How to Subtract Mixed Fractions from Whole Numbers?

The whole numbers can be modified and be written as a mixed fraction. Once the whole number is written in the form of a mixed fraction, the general steps of subtracting the mixed fractions can be followed.
For example, let's perform 5 - \(2\dfrac{2}{3}\)
Note that 5 can also be written as 4 + 1 = 4 + (3/3) = \(4\dfrac{3}{3}\)
Thus, we will perform \(4\dfrac{3}{3}\) - \(2\dfrac{2}{3}\)
= (4 - 2) + (3/3) - (2/3)
= 2 + (1/3)
= \(2\dfrac{1}{3}\)

How to do Adding and Subtracting Mixed Fractions?

Mixed fractions will be subtracted from mixed fractions by first converting them into improper fractions and subtracting their numerators if they have the same denominator. If they have different denominators, then they are first converted into the same denominators by taking their LCM followed by subtracting their numerators. The final result will be converted back to a mixed fraction. The steps to add mixed fractions also remain the same. The only difference is we add the numerators instead of subtracting.
For example, let's perform \(4\dfrac{2}{7}\) - \(3\dfrac{1}{6}\)
= (30/7) - (19/6)
= [(30 × 6) / (7 × 6)] - [(19 × 7) / (6 × 7)]
= (180/42) - (133/42)
= 47/42
= \(1\dfrac{5}{42}\)