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208 LearnersLast updated on December 8, 2025
Multiplying Mixed Fractions
A mixed fraction is a combination of a whole number and a proper fraction. Multiplication of mixed fractions is the multiplication of two mixed fractions. In this article, we will learn about the multiplication of mixed fractions.
What are Mixed Numbers?
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.
Examples:
- \(1\frac{1}{2}\)
- \(3\frac{3}{4}\)
- \(5\frac{2}{5}\)
- \(2\frac{1}{3}\)
- \(10\frac{1}{8}\)
How to Convert 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.
Difference Between Proper, Improper and Mixed Fractions
| Proper Fractions | Improper Fractions | Mixed Fractions |
| Numerator less than Denominator. | The numerator is greater than or equal to the denominator. | A combination of a whole number and a proper fraction. |
| The value is 0 < 1. | The value is 1 ≤ a. | The value is greater than 1. |
| Example: \(3 \over 4\), \(2 \over 3\) | Example: \(3 \over 2\), \(13 \over 6\) | Example: \(5{2 \over 3}\), \(7{4 \over 5}\) |
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What is Multiplying Mixed Numbers?
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
Multiplying mixed fractions with like denominators is the process of multiplying two mixed fractions that share the same bottom number. Follow these steps to multiply the mixed fractions with like denominators:
The Steps
- Convert to Improper Fractions: Change each mixed number into an improper fraction.
- Multiply the Numerators: Multiply the top numbers together.
- Multiply the Denominators: Multiply the bottom numbers together (Note: Do not keep the denominator the same).
- Simplify: Convert the result back to a mixed number or simplify if possible.
Example:
\(1\frac{2}{5} \times 2\frac{1}{5}\)
- Step 1: Convert to improper fractions.
\(1\frac{2}{5} = \frac{7}{5}\ and\ 2\frac{1}{5} = \frac{11}{5}\)
- Step 2 & 3: Multiply numerators and denominators.
\(\frac{7 \times 11}{5 \times 5} = \frac{77}{25}\)
- Step 4: Convert back to a mixed number.
\(77 \div 25 = 3\)
with a remainder of 2
Answer:
\(3\frac{2}{25}\)
Multiplying Mixed Fractions with Unlike Denominators
Multiplying mixed fractions with unlike denominators is the process of multiplying two mixed fractions that have different bottom numbers. Follow these steps to multiply the mixed fractions with unlike denominators:
The Steps
- Convert to Improper Fractions: Change each mixed number into an improper fraction.
- Multiply the Numerators: Multiply the top numbers together.
- Multiply the Denominators: Multiply the bottom numbers together (Note: You do not need to find a common denominator for multiplication).
- Simplify: Convert the result back to a mixed number or simplify if possible.
Example:
\(1\frac{1}{2} \times 1\frac{2}{5}\)
- Step 1: Convert to improper fractions.
\(1\frac{1}{2} = \frac{3}{2}\ and\ 1\frac{2}{5} = \frac{7}{5}\)
- Step 2 & 3: Multiply numerators and denominators.
\(\frac{3 \times 7}{2 \times 5} = \frac{21}{10}\)
- Step 4: Convert back to a mixed number.
\(21 \div 10 = 2\)
with a remainder of 1
Answer:
\(2\frac{1}{10}\)
Multiplying Mixed Fractions and Proper Fractions
Multiplying mixed fractions and proper fractions is the process of multiplying a mixed number by a fraction where the numerator is less than the denominator. Follow these steps to multiply mixed fractions and proper fractions:
The Steps
- Convert to Improper Fraction: Change the mixed number into an improper fraction. Keep the proper fraction as it is.
- Multiply the Numerators: Multiply the top numbers together.
- Multiply the Denominators: Multiply the bottom numbers together.
- Simplify: Reduce the fraction to its lowest terms or convert it back to a mixed number if the result is improper.
Example
\(2\frac{2}{5} \times \frac{1}{3}\)
- Step 1: Convert the mixed fraction to an improper fraction.
\(2\frac{2}{5} = \frac{12}{5}\)
- Step 2 & 3: Multiply numerators and denominators.
\(\frac{12 \times 1}{5 \times 3} = \frac{12}{15} \)
- Step 4: Simplify the fraction (divide both numbers by 3).
\(12 \div 3 = 4\)
\(15 \div 3 = 5\)
Answer:
\(\frac{4}{5}\)
Multiplying Mixed Fractions with Whole Numbers
Multiplying mixed fractions with whole numbers is the process of multiplying a mixed number by a standard integer. Follow these steps to multiply mixed fractions with whole numbers:
The Steps
- Convert to Improper Fraction: Change the mixed number into an improper fraction.
- Convert Whole Number to Fraction: Write the whole number as a fraction by placing it over 1.
- Multiply the Numerators: Multiply the top numbers together.
- Multiply the Denominators: Multiply the bottom numbers together.
- Simplify: Convert the result back to a mixed number or simplify if possible.
Example
\(3 \times 1\frac{1}{2}\)
- Step 1: Convert the mixed fraction to an improper fraction.
\(1\frac{1}{2} = \frac{3}{2}\)
- Step 2: Convert the whole number to a fraction.
\(3 = \frac{3}{1}\)
- Step 3 & 4: Multiply numerators and denominators.
\(\frac{3 \times 3}{1 \times 2} = \frac{9}{2}\)
- Step 5: Convert back to a mixed number.
\(9 \div 2 = 4\)
with a remainder of 1
Answer:
\(4\frac{1}{2}\)
Tips and Tricks to Master Multiplying Mixed Fractions
Getting comfortable with what are mixed numbers and how they behave during multiplication can be a bit of a hurdle. It’s really common for students to just want to multiply the big numbers and the fractions separately—it feels intuitive, but unfortunately, it gives the wrong answer! To help clear up the confusion around multiplication of mixed numbers, here are some friendly strategies that really stick:
- Draw it Out with Area Models: Sometimes, seeing is believing. To show how to multiply mixed fractions without just memorizing rules, try sketching a rectangle (an area model). Split the sides into the whole number and the fraction. It visually proves that there are actually four parts to multiply, not just two. It’s a great "aha!" moment for why we can't take shortcuts.
- Get "MAD" at Fractions: This is a fun memory aid that students love. To convert a mixed number into an improper fraction, tell them to get "MAD": Multiply the whole number by the bottom, Add the top, and keep the Denominator the same. It’s a catchy little routine that gets them ready for mixed fraction multiplication without the stress.
- The "Ballpark" Estimation: Before doing the heavy lifting, take a second to guess the answer. Round each mixed number to the nearest whole number and multiply them in your head. It gives you a "ballpark" figure. If the final answer for the mixed numbers multiplication is miles away from that guess, it’s a great signal that something went wrong in the calculation.
- The Golden Rule – Go Improper First: If there is one rule to live by, it's this: for how to multiply fractions with mixed numbers, you must turn them into improper fractions first. It’s non-negotiable! Remind learners that while they might get away with shortcuts in addition, multiplication changes the structure of the numbers entirely.
- Simplify Early to Save Headaches: Once you convert to improper fractions, the top numbers can get huge and scary. Encourage students to look for common numbers to cross-cancel before they multiply mixed fractions. It keeps the numbers small and manageable, meaning there is less chance of making a silly mistake at the end.
- Make it Real: Let’s be honest, abstract math can be dry. Bring the concept to life by talking about mixed fraction multiplication in real scenarios. Whether it’s doubling a cookie recipe (using \(2\frac{1}{2}\) cups of flour) or measuring a room for a new carpet, giving the numbers a job to do helps students understand why the values change.
- Mix Up the Practice: Doing sheet after sheet of problems can get boring fast. Keep the energy up by using different tools—maybe a game, a puzzle, or a fun mixed fraction multiplication worksheet with word problems. When students see the concept of mixed numbers in different formats, it stops being rote memorization and starts becoming real understanding.
Common Mistakes and How to Avoid Them in Multiplying Mixed Fractions
When multiplying mixed fractions, students tend to make mistakes. Here are some common mistakes and ways to avoid them.
Mistake 1
Skipping the final simplification step
The result was left without simplifying it from improper to mixed fractions. Always check if the fraction can be simplified or turned into a mixed fraction after multiplying.
Mistake 2
Not simplifying large numbers before the final step
Students often do not express the final answer as a mixed number or reduce it fully. Check if the final answer can be reduced further, and simplify it. If necessary, convert it to a mixed number.
Mistake 3
Multiplying fractions without simplifying the expressions
Direct multiplication of large numbers without simplification, which causes more difficulty in calculations. To check if we can cross-cancel the expressions before multiplying.
Mistake 4
Applying addition or division where multiplication is required.
Causing errors in calculation, especially when dealing with mixed numbers. Keeping yourself updated with the methods and rules, making sure the right operation is used.
Mistake 5
While converting the mixed number, students may misplace the whole number.
Converting a mixed number into an incorrect, improper fraction without using the right steps or formula. Using the correct formula for solving,
(Whole number × Denominator) + Numerator = New Numerator.
Real-Life Applications of Multiplying Mixed Fractions
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.
Solved Examples for Multiplication of Mixed Fractions
Problem 1
Multiply 4 5/6 and 3 1/2
\(16 \frac{11}{12} \)
Explanation
Given, \(4 \frac{5}{6} \) × \(3 \frac{1}{2} \)
Let us convert the given mixed fractions into improper fraction.
\(4 \frac{5}{6} \) = \(4 \times 6 + 5 = 24 + 5 = 29 = \frac{29}{6} \)
\(3 \frac{1}{2} \) = \(3 \times 2 + 1 = 6 + 1 = 7 = \frac{7}{2} \)
Now, let's multiply the fractions.
\(\frac{29}{6} \times \frac{7}{2} = \frac{29 \times 7}{6 \times 2} = \frac{203}{12} \)
Convert it into mixed fraction.
\(\frac{203}{12} = 16 \tfrac{11}{12} \)
Problem 2
Multiply 6 3/2 and 3
\(22 \frac{1}{2} \)
Explanation
Given, \(6 \frac{3}{2} \times 3 \)
Let us convert the given mixed fraction to improper fraction.
\(6 \tfrac{3}{2} = 6 \times 2 + 3 = 12 + 3 = 15 = \frac{15}{2} \)
\(3 = \frac{3}{1} \)
Multiply the two fractions
\(\frac{15}{2} \times \frac{3}{1} = \frac{15 \times 3}{2 \times 1} = \frac{45}{2} \)
Convert it to mixed fraction.
\(\frac{45}{2} = 22 \tfrac{1}{2} \)
Problem 3
Multiply 5 1/3 and 3/2
8
Explanation
Given \(5 \tfrac{1}{3} \times \frac{3}{2} \)
Let us convert the given mixed fraction to improper fraction.
\(5 \tfrac{1}{3} = 5 \times 3 + 1 = 15 + 1 = 16 = \frac{16}{3} \)
\(\frac{16}{3} \times \frac{3}{2} = \frac{16 \times 3}{3 \times 2} \)
\(\frac{16 \times 3}{3 \times 2} = \frac{48}{6} \)
Upon simplification,
\(\frac{48}{6} = 8 \)
Problem 4
If it takes 1 1/4 liters of water to water a plant, calculate how much water we need to water 6 plants.
\(7 \frac{1}{2} \)
Explanation
Each plant needs \(1 \frac{1}{4} \) liters of water.
In order to water 6 plants, we need to multiply \(1 \frac{1}{4} \) with 6
Let us convert the mixed fraction
\(1 \tfrac{1}{4} = \frac{5}{4} \)
Let's multiply the two values.
\(\frac{5}{4} \times 6 = \frac{30}{4} \)
\(\frac{30}{4} = 7 \tfrac{1}{2} \)
Problem 5
If we want 3 1/2 kg of apples for each of your 5 friends, then how much should we buy?
\(17 \frac{1}{2} \)
Explanation
Let's convert the mixed fraction
\(3 \tfrac{1}{2} = \frac{7}{2} \)
Now multiply the numbers, to get the total apples we need to buy.
\(3 \tfrac{1}{2} \times 5 = \frac{7}{2} \times 5 = \frac{35}{2} \)
Convert it into mixed fraction
\(\frac{35}{2} = 17 \tfrac{1}{2} \)
FAQs on Multiplying Mixed Fractions
1.How to multiply mixed fractions?
2.When multiplying mixed fractions, should we find a common denominator?
3.If my final answer is an improper fraction, what should I do?
4.Can I cross-cancel or simplify the expression before multiplication?
5.What are the real-life applications of multiplying mixed fractions?
6.How can I help my child understand it better?
7.How often should my child practice multiplying mixed fractions?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
