Multiplying Mixed Numbers – Definition with Examples

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What Is Multiplying Mixed Numbers?

Multiplying mixed numbers refers to finding the product of two mixed numbers. Before delving into the steps and examples, let’s take a moment to review some essential basics.

A fraction represents parts of a whole. If a cake is divided into eight equal pieces, and three pieces of the cake are placed on a plate, then we can say that the plate has 38 of the cake. 

There are three types of fractions:

Proper FractionsImproper FractionsMixed Numbers(Mixed Fractions)
Numerator < DenominatorNumerator ≥ DenominatorConsists of a whole number and a proper fraction
They lie between 0 and 1.1 or greater than 1 1 or greater than 1
Examples: 12,34,712Examples: 32,125,83Examples: 312;223;217

Let’s explore mixed numbers.

What Are Mixed Numbers?

A mixed number or a mixed fraction consists of a whole number and a proper fraction. It generally represents a number between any two whole numbers. 

Look at the given image. It represents a fraction that is greater than 1 but less than 2. It is a mixed number. 

Visual representation of a mixed number

Some other examples of mixed numbers are: 

Examples of mixed numbers
  • Parts of a Mixed Number

A mixed number is formed by combining three parts: a whole number, a numerator, and a denominator. The numerator and denominator are part of the proper fraction that makes the mixed number.

Parts of a mixed number

How to Multiply Fractions

Before we discuss how to multiply mixed numbers, let’s quickly understand how to multiply two fractions.

Multiplication of two fractions is a pretty straightforward process. To multiply two fractions ab and cd, we first multiply the numerators and write the product as the numerator of the answer. Next, we multiply the denominators and write the result in the denominator. This is how the multiplication of fractions works. The fractions can be proper or improper.

ab×cd=a×cb×d

Example: 23×79=2×73×9=1427

How to Convert Mixed Numbers into Improper Fractions

Converting mixed numbers into improper fractions simplifies the process of multiplying two mixed numbers or multiplying a mixed number and a fraction. Let’s understand the steps.

  • Step 1: Multiply the whole number by the denominator of the fraction.
  • Step 2: Add the answer obtained from Step 1 to the numerator of the fraction. 
  • Step 3: Write an answer obtained from step 2 over the denominator.

Example 1: Suppose we have to convert 223 into an improper fraction.

Step 1: Multiply 3 and 2. We get 3×2=6.

Step 2: Add 6 and 2. We get 6+2=8

Step 3: The fraction obtained is 83

In summary, 223=(3×2)+23=83

Example 2:  213=(3×2)+13=73

Example 3: 956=(6×9)+56=596

How to Multiply Mixed Numbers

The simplest way to multiply two mixed numbers is to convert mixed numbers into improper fractions and then multiply the resulting improper fractions using the usual fraction multiplication rules. 

We will discuss 3 different cases here:

  • Multiplying mixed numbers with whole numbers
  • Multiplying mixed fractions 
  • Multiplying mixed numbers with fractions

How to Multiply a Mixed Number and a Whole Number

Step 1: Convert the mixed number into an improper fraction.

Step 2: Rewrite the whole number as a fraction with the denominator 1.

Step 3: Multiply two fractions by multiplying the numerators and denominators separately.

Step 4: Convert it into simplified form if required.

Example: Multiply 3 and 212.

212=(2×2)+12=52

352=31×52=152=712

How to Multiply Two Mixed Numbers

Step 1: Convert the mixed numbers into improper fractions.

Step 2: Multiply the two fractions by multiplying the numerators and denominators separately.

Step 3: Convert it into simplified form if required.

Example: Multiply 412 and 313.

412=4×2+12=92

313=3×3+13=103

412×313=92×103=906=15 

How to Multiply a Mixed Number and a Fraction

Multiplying fractions with ​​mixed numbers refers to finding the product of a fraction and a mixed number (mixed fraction). 

The first step in multiplying fractions and mixed numbers is to convert the mixed number into an improper fraction. After that, you proceed with the regular multiplication of two fractions.

Here are the steps for multiplying fractions with mixed numbers:

Step 1: Convert the mixed number into an improper fraction.

Step 2: Multiply the numerators of the fractions separately. Multiply the denominators of the fractions separately.

Step 3: Convert the resulting fraction into its simplified form if required.

Example 1: Multiply 23 with 215.

Step 1: Convert all the mixed numbers in the given problem into improper fractions.

Here, 215 is a mixed number. Converting it into an improper fraction, we get

215=(5×2)+15=115

Step 2: Rewrite the problem using the new improper fraction.

Here, the problem becomes 

23×215=23×115

Step 3: Multiply the numerators. Write the result as the numerator of the answer.

Multiply the denominators. Write the result as the denominator of the answer. 

23×115=2×113×5=2215

Step 4: Write the fraction in its simplest form. Rewrite the answer as a mixed number if required. 

Here, 2215=1715

Example 2: Multiply 25 and 312.

312=(3×2)+12=72

25×72=1410=75=125

Different Cases: We might encounter three cases when multiplying a fraction with a mixed number. 

  • Multiplying a proper fraction with a mixed number
  • Multiplying a proper fraction with a mixed number
  • Multiplying a mixed fraction with a mixed fraction

In each case, we follow the same steps we discussed earlier. Let’s see examples in every category. 

Example 1: Multiplying a proper fraction with a mixed number 

Multiply 15 by 212.

Convert the mixed number into an improper fraction. 

212=(2×2)+12=52

Next, we multiply the fractions 15 and 52.

15×52=1×55×2=12

Example 2: Multiplying an improper fraction with a mixed number

Multiply 98 by 613. 

Convert the mixed number into an improper fraction. 

613=(6×3)+13=193

Next, we multiply the fractions 98 and 193.

9×198×3=578

Change the improper fraction 578 into a mixed number. 

578=718

So, 98×613=718

Example 3: Multiplying two mixed numbers

Multiply 315 by 623

Convert both the mixed numbers to improper fractions.

315=(3×5)+15=165 

623=(6×3)+23=203

Multiply the fractions 165 and 203.

16×205×3=32015

Change the improper fraction 32015 into a mixed number. 

32015=21515

So, 315×623=19515

Conclusion

In this article, we learned how to multiply mixed numbers. We started with an overview of the basic essential concepts, such as fractions, mixed numbers, multiplying fractions, and converting mixed numbers into improper fractions. We discussed different cases involved with various examples. Now, let’s dive into some solved examples and practice MCQ problems. Happy solving!

Solved Examples on Multiplying Mixed Numbers

1. Multiply 537 by the multiplicative inverse of 735

Solution:

537=5×7+37=387

735=7×5+35=385

Multiplicative inverse of 385 is 538.

Product =387×538=57

2. Emma walks 523 miles in a day. How much distance will she cover in 9 days?

Solution: 

Distance traveled by Emma in 1 day =523 miles =173 miles

Distance traveled by Emma in 9 days =9×173=51 miles 

3. Find 625×34

Solution: 

625=6×5+25=325

325×34=32×35×4=9620=245=445

4. Multiply fractions with mixed numbers and simplify.

i) 19×214

ii) 25×137

Solution: 

i) Convert the mixed number 214 into an improper fraction.

214=(2×4)+14=94

Multiply 94 with 19.

19×94=14

Thus, 19×214=14

ii) Convert the mixed number 137 into an improper fraction.

137=107

Multiply 25 with 107.

25×107=47

Thus, 25×137=47

5. What is the product of 53 and 825

Solution:

Convert the mixed number 825 into an improper fraction.

825=(8×5)+25=425

Multiply 53 with 425.

53×425=14.

Thus, 53×825=14

6. What is the result when 225 is multiplied by 113?

Solution:

Convert both the mixed numbers to improper fractions.

225=(2×5)+25=125

113=(1×3)+13=43

Multiply 125 with 43.

125×43=4815=165=315

Thus, 225×113=315

Practice Problems on Multiplying Mixed Numbers

Multiplying Mixed Numbers - Definition with Examples

Attend this quiz & Test your knowledge.

1

Which of these is the first step to multiply mixed numbers?

Multiply whole numbers
Multiplying the numerators
Multiplying the denominators
Converting the mixed numbers into improper fractions
CorrectIncorrect
Correct answer is: Converting the mixed numbers into improper fractions
Converting the mixed numbers to improper fractions The first step to multiplying mixed numbers is to convert them into improper fractions.
2

On multiplying 1016 by 2211, we get ____.

whole number
mixed number
proper fraction
negative number
CorrectIncorrect
Correct answer is: mixed number
1016×2211=616×2411=24411=22211, i.e., a mixed number
3

The value of 429×117 is:

15263
25263
45263
6163
CorrectIncorrect
Correct answer is: 45263
429×117=389×87=30463=45263
4

On multiplying 56 by 212, we get ____.

216
2112
3112
116
CorrectIncorrect
Correct answer is: 2112
212=52
56×52=2512=2112
5

389×127=

123
212
4
5
CorrectIncorrect
Correct answer is: 5
389=3×9+89=359 and 127=(1×7)+27=97
359×97=5
6

To multiply mixed numbers with fractions, we convert mixed numbers into

decimals
improper fractions
whole numbers
proper fractions
CorrectIncorrect
Correct answer is: improper fractions
For multiplying mixed numbers with fractions, we convert the mixed number into an improper fraction.
7

423×217=?

10
12
15
20
CorrectIncorrect
Correct answer is: 10
423=(4×3)+23=143
217=157
157×143=14×157×3=10

Frequently Asked Questions about Multiplying Mixed Numbers

No. The denominators need not be equal to multiply two fractions or two mixed numbers.

The other name for mixed numbers is mixed fractions.

No. A mixed number is always greater than 1. So, the product of 2 numbers greater than 1 will always be greater than 1, i.e., a mixed/whole number.

To multiply a fraction ab with a whole number m, we express the whole number in fractional form as m1. Next, we multiply the two fractions ab and m1. For example, 25×81=165

We first convert the mixed number to an improper fraction for dividing fractions with mixed numbers. Next, to divide the fraction with the improper fraction, we multiply the given fraction by the reciprocal of the improper fraction.

23÷217=23÷157=23×715=1445