Multiplying Mixed Fractions

The operation of multiplication between any two mixed fractions is known as multiplying mixed fractions. Mixed fractions can be defined as a fraction containing a whole number and a proper fraction. It is also another way to represent an improper fraction. We will be studying more about the steps and examples of multiplying mixed fractions in this article.

Multiplying mixed fractions with like denominators is defined as the multiplication operation between two mixed fractions of the same denominator. Look at the following points to be kept in mind while multiplying 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 \(1\dfrac{3}{5}\) to an improper fraction, we multiply 1 and 5 i.e, 1 × 5 = 5 and the result is added to 3 i.e., 5 + 3 = 8. Thus the improper fraction is 8/5.
• 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 17/5 to a mixed number, we first divide 17 by 5 and get the quotient as 3 and the remainder as 2. Thus, the mixed number is \(3\dfrac{2}{5}\).

Let's take an example to understand multiplying mixed fractions with like denominators.

Example: Multiply the mixed fractions \(2\dfrac{2}{7}\) and \(3\dfrac{1}{7}\).

• Step 1: Convert the given mixed fractions \(2\dfrac{2}{7}\) and \(3\dfrac{1}{7}\) to improper fractions. i.e., \(2\dfrac{2}{7}\) = 16/7 and \(3\dfrac{1}{7}\) = 22/7.
• Step 2: The numerators and denominators of these fractions are multiplied separately. i.e., (16 × 22) / (7 × 7).
• Step 3: Cancel out the common factors if they exist in the numerator and the denominator. In this example, there are no common factors.
• Step 4: Perform the calculation. (16 × 22) / (7 × 7) = 352/49.
• Step 5: If the result obtained in the previous step is an improper fraction, convert it to a mixed fraction. i.e., 352/49 = \(7\dfrac{9}{49}\).

Multiplying mixed fractions with unlike denominators is defined as the multiplication between two mixed fractions having different denominators such as \(1\dfrac{2}{9}\) and \(2\dfrac{1}{4}\). The steps to multiply the mixed fractions remain the same irrespective of like or unlike denominators as discussed in the previous section. Let us take an example of multiplying mixed fractions to understand it better.

Example: Lets's multiply two mixed fractions \(1\dfrac{2}{9}\) and \(2\dfrac{1}{4}\) that have unlike denominators.

\(1\dfrac{2}{9}\) × \(2\dfrac{1}{4}\)

= (11/9) × (9/4) [On converting them to improper fractions]

= (11 × 9) / (9 × 4)

= 11/4 [By canceling out the common factors]

= \(2\dfrac{3}{4}\) [By converting to a mixed fraction]

Thus, the value of \(1\dfrac{2}{9}\) × \(2\dfrac{1}{4}\) is \(2\dfrac{3}{4}\).

Let's take an example to understand the steps of multiplying mixed fractions and proper fractions.

Example: Multiply the fraction 1/5 with \(3\dfrac{1}{3}\).

We have to perform (1/5) × \(3\dfrac{1}{3}\).

• Step 1: We will convert the given mixed fraction to an improper fraction. i.e., \(3\dfrac{1}{3}\) = 10/3.
• Step 2: We will now multiply the given fraction 1/5 with the fraction obtained in the previous step. i.e., (1/5) × (10/3).
• Step 3: To multiply (1/5) × (10/3) we will multiply the numerators and the denominators of the fractions. i.e., (1 × 10) / (5 × 3).
• Step 4: To simplify this, we will cancel out the common factors in the numerator and denominator that will give us 2/3.
• Step 5: After performing the calculation, we get 2/3 as the result.
• Step 6: If the result in the previous step is an improper fraction, we convert it back to a mixed fraction.

The operands in multiplying mixed fractions with whole numbers are a mixed fraction and a whole number being multiplied. The whole number can be written in a fraction format assuming the denominator to be 1. For example, 5 can be written as 5/1. The steps of multiplying mixed fractions with whole numbers will be very similar to the previous section. Let's take an example to understand this.

Example: Multiply the mixed fraction \(2\dfrac{1}{6}\) with a whole 3.

\(2\dfrac{1}{6}\) × 3

= (13/6) × (3/1) [On converting the mixed fraction to improper fraction and writing the whole number in terms of fraction]

= (13 × 3) / (6 × 1)

= 13/2 [Cancelling out the common factors and calculating]

= \(6\dfrac{1}{2}\) [Converting the result back to a mixed fraction]

This is how we do the multiplication of mixed fractions with whole numbers.

Related Articles on Multiplying Mixed Fractions

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

• Mixed Fractions
• Improper fractions
• Proper fraction
• Fractions

How to Solve Multiplying Mixed Fractions?

Multiplying mixed fractions is done by first converting them into an improper fraction followed by multiplying their corresponding numerators and denominators separately and simplifying.
For example, \(3\dfrac{5}{7}\) × \(1\dfrac{2}{3}\)
= (26/7) × (5/3)
= (26 × 5) / (7 × 3)
= 130/21
= \(6\dfrac{4}{21}\)

How to Multiply Mixed Fractions with Whole Numbers?

For multiplying mixed fractions with whole numbers we will write the whole number in fractional form by writing the denominator as 1 and changing the mixed fraction to an improper fraction followed by multiplying the two fractions. For example, \(1\dfrac{2}{3}\) × (7/1)
= (5/3) × (7/1)
= (5 × 7) / (3 × 1)
= 35/3 = \(11\dfrac{2}{3}\)

How to Multiply Mixed Fractions with Improper Fractions?

Mixed fractions are first converted into an improper fraction followed by multiplying the result with the given improper fraction. The final result is simplified and converted back to a mixed fraction if it's an improper fraction.
For example, \(2\dfrac{1}{8}\) × (26/17)
= (17/8) × (26/17)
= 13/4
= \(3\dfrac{1}{4}\)

How to Multiply Mixed Fractions with the Same Denominators?

Multiplying mixed fractions with the same denominators are done by first converting the mixed fractions to improper followed by multiplying the numerators and denominators separately and simplifying it to get the result.
For example, \(4\dfrac{2}{3}\)× \(3\dfrac{1}{3}\)
= (14/3) × (10/3)
= (14 × 10) / (3 × 3)
= 140/9
= \(15\dfrac{5}{9}\)

How to Multiply Mixed Fractions with Different Denominators?

Multiplying mixed fractions with different denominators can easily be multiplied just like regular multiplication of mixed fractions. For this, the mixed fractions will be converted to an improper fraction initially. Next, the numerator of these improper fractions will be multiplied. The denominators will also be multiplied. The result will be further simplified to obtain the result.
For example, \(3\dfrac{1}{4}\) × \(3\dfrac{1}{3}\)
= (13/4) × (10/3)
= (13 × 10) / (4 × 3)
= (13 × 5) / (2 × 3)
= 65/6
= \(10\dfrac{5}{6}\)

How to Multiply Mixed Fractions with Proper Fractions?

To multiply mixed fractions with proper fractions, we first convert the mixed fraction to an improper fraction followed by multiplying the two fractions and simplifying them.
For example, let's multiply 7/8 and \(1\dfrac{4}{7}\)
= (7/8) × \(1\dfrac{4}{7}\)
= (7/8) × (11/7)
= (7 × 11) / (8 × 7)
= 11/8