Multiplying Mixed Fractions

Mixed numbers are a combination of a whole number and a proper fraction. For example, \(5 \frac{1}{3} \), here 5 is the whole number and \(\frac{1}{3} \) is the fractional part of the mixed number. To perform basic operations using mixed fractions, we convert the mixed numbers to improper fractions. 

Conversion of mixed fractions to improper fractions includes multiplication and addition. The steps are explained below

1. Firstly, the denominator is multiplied by the whole number.

2. The result of the first step has to be added to the numerator

3. Then, the sum is written over the original denominator.

Example: Convert \(5 \frac{2}{3} \) into an improper fraction.

Solution: \(5 \times 3 + 2 = 15 + 2 = \frac{17}{3} \)

\(\frac{17}{3} \) is the improper fraction of the given mixed fraction.

Multiplication is a basic mathematical operation that helps find the product of numbers. Finding the product of two mixed numbers is multiplying mixed numbers. 

Example: Multiply \(5 \frac{2}{3} \) and  \(2 \frac{1}{2} \)

Solution: Converting the given mixed fractions to improper fractions,
\(5 \frac{2}{3} \) = \(5 \times 3 + 2 = 15 + 2 = 17 = \frac{17}{3} \)
\(2 \frac{1}{2} \) = \(2 \times 2 + 1 = 4 + 1 = 5 = \frac{5}{2} \)

As we converted the mixed numbers to improper fractions, now we will multiply the improper fractions,
\(\frac{17}{3} \times \frac{5}{2} = \frac{85}{6} \)
Converting this back to a mixed number: 
\(\frac{85}{6} = 14 \tfrac{1}{6} \)

Multiplying mixed fractions with like denominators is the process of multiplying two mixed fractions with same denominators. Follow these steps to multiply the mixed fractions with like denominators:
 

• Mixed fractions are converted to improper fractions
   
• Multiply the numerators and the denominators
   
• If necessary, convert the result back to a mixed fraction.
   

Example: Multiply \(3 \frac{3}{2} \) and \(4 \frac{5}{2} \)

Solution: Converting the given mixed fractions to improper fractions,
 \(3 \frac{3}{2} \) = \((3 \times 2) + 3 = \frac{9}{2} \)
\(4 \frac{5}{2} \) = \((4 \times 2) + 5 = \frac{13}{2} \)
Next is to multiply the improper fractions,
\(\frac{9}{2} \times \frac{13}{2} = \frac{9 \times 13}{2 \times 2} = \frac{117}{4} \)
Converting this back to a mixed number,
\(\frac{117}{4} = 29 \tfrac{1}{4} \)

The multiplication of mixed fractions with unlike denominators is the multiplication of two mixed fractions with different denominators. In this section, we learn how to multiply it step-by-step with an example.

Multiplying \(2 \frac{1}{2} \) and  \(3 \frac{2}{5} \)

Step 1:  Converting the given mixed fractions to improper fractions,
\(2 \frac{1}{2} \) = \(2 \times 2 + 1 = 4 + 1 = 5 = \frac{5}{2} \)
\(3 \frac{2}{5} \) = \(3 \times 5 + 2 = 15 + 2 = 17 = \frac{17}{5} \)

Step 2: Next, multiply the improper fractions,
\(\frac{5}{2} \times \frac{17}{5} = \frac{5 \times 17}{2 \times 5} = \frac{85}{10} \)

Step 3: Converting the result back to a mixed number,
 \(\frac{85}{10} \) =  \(8 \frac{5}{10} \) = \(8 \frac{1}{2} \)

Now we learned how to multiply mixed numbers with like and unlike fractions. Now let’s see how to multiply mixed fractions and proper fractions. Here are the steps to multiply mixed fractions and proper fractions with an example,

Example: Multiply \(4 \frac{5}{2} \) and \(\frac{2}{3} \)

Step 1:  Converting the given mixed fraction to an improper fraction,
\(4 \frac{5}{2} \) = \((4 \times 2 + 5) = \frac{13}{2} \)

Step 2: Next, multiply the improper fractions and the proper fraction.
\(\frac{13}{2} \times \frac{2}{3} = \frac{13 \times 2}{2 \times 3} = \frac{26}{6} \)

Step 3: Converting the result back to a mixed number,
\(\frac{26}{6} \) = \(4 \frac{2}{6} \) = \(4 \frac{1}{3} \)

We can even multiply a mixed fraction by a whole number. To multiply mixed fractions with whole numbers, follow the steps given below with an example.

For example, multiply \(5 \frac{3}{2} \) and 4.

Step 1:  Converting the given mixed fraction to an improper fraction,
To convert, we first multiply the whole number with the denominator and then add the product to the numerator. Then the sum will be written on a numerator with the original denominator. 
\(5 \frac{3}{2} \) = \((5 \times 2) + 3 = 10 + 3 = 13 = \frac{13}{2} \)

Step 2: Write the whole number as a fraction,
\(4 = \frac{4}{1} \)

Step 3: Next, multiplying the fractions, we get
\(\frac{13}{2} \times \frac{4}{1} = \frac{4}{1} = \frac{52}{2} \)

Step 4: Converting the result back to a mixed number,
\(\frac{52}{2} = 26 \).

Many students find multiplying mixed fractions tricky. By following these tips and tricks, students can master multiplying mixed fractions.
 

• Always convert the mixed numbers to improper fractions before performing mathematical operations. 
   
• Parents can help their kids by using fraction strips or pie models. Draw some diagrams to show multiplication parts of a whole.
   
• Simplify the fraction before multiplying to make the calculations easier. For example, when multiplying \(\frac{8}{9} \) and \(\frac{3}{4} \), 8 and 4 have a common factor of 4, and 3 and 9 have a common factor of 3. So, it can be simplified as \(\frac{2}{3} \times \frac{1}{1} \). 
   
• Use mnemonics and memory tricks to convert, multiply, simplify and convert back for faster solving of problems.
   
• Make learning process visual and interactive by using apps or online games focused on fraction multiplication.

Let's explore about some real-life situations where whole numbers and fractions appear together, which is exactly what mixed fractions represent. 
 

• Cooking and baking: Cooking recipes often require measurements in fractions. We may have to add \(2 \frac{1}{2} \) cups of sugar to make a cake. If we have to make 3 cakes, we multiply these numbers to get \(\frac{15}{2} \) cups of sugar.
   
• Construction and carpentry: Measurement of wood, metal, or fabric often uses mixed fractions. We may have to cut 4 planks of length \(2 \frac{3}{4} \) each. We multiply these digits to get 11 planks.
   
• Time scheduling: We can calculate the total times a task can be repeated. We can calculate the time period it'll take to finish a work.
   
• Travel and distance: For distances that are in mixed fractions, we can use the multiplication of mixed fractions to calculate the total distance traveled in a certain period of time.
   
• Gardening and landscaping: For planting, watering and layering materials repeatedly, we can use the multiplication properties of mixed fractions when the quantities are in mixed fraction.