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The road to delicious cakes is paved with mixed numbers.
How?
Think about the last recipe you saw. It probably called for “\(\large 2\frac{1}{2}\) cups of flour” or “\(\large 1\frac{3}{4}\) teaspoons of vanilla”.
Chances are, you’ve worked with mixed numbers long before ever seeing them in your math book!
Whether you're just learning about them or reviewing what you already know, this guide will help you truly understand mixed numbers.
Read on to find clear definitions, examples of operations with mixed numbers, and answers to questions students usually ask.
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Imagine you have three apples and the fourth one that you cut into four slices. Your friend comes over and eats one slice, so now you have three-quarters of the fourth apple.
How many apples do you have in total?
You have \(\large 3\frac{3}{4}\) apples—more than 3, but not quite 4. That’s called a mixed number!
A mixed number is a type of fraction made up of:
A whole number (like the 3 in \(\large 3\frac{3}{4}\))
A fraction (like the \(\large \frac{3}{4}\) in \(\large 3\frac{3}{4}\))
Let’s break it down once more:
The whole number represents complete objects such as 3 apples.
The fraction represents a part of another whole such as \(\large \frac{3}{4}\) of an apple.
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Now that we know what mixed numbers are, let’s explore what we can do with them!
Just like whole numbers and fractions, mixed numbers can be converted, added, subtracted, multiplied, and divided.
Each of these operations has its own steps, and we’ll go through them one at a time.
Sometimes, a fraction has a numerator (the top number) that is larger than the denominator (the bottom number) —like \(\large \frac{9}{4}\) or \(\large \frac{7}{3}\). This is called an improper fraction because it represents more than one whole.
But when we read \(\large \frac{9}{4}\), it doesn’t clearly tell us how many whole things we have, right?
That’s where mixed numbers come in! They break things down into whole parts and a leftover fraction and make it easier to picture amounts, compare values, and use them in real life.
That’s why we convert improper fractions into mixed numbers when we need a clearer answer.
And how do we do that?
At Mathnasium, we love to show principles through examples—so let’s walk through one!
Let’s convert \(\large \frac{7}{2}\) into a mixed number.
Divide the numerator (7) by the denominator (2):
7 ÷ 2 = 3, with remainder 1
This tells us that 2 fits into 7 a total of 3 times, with 1 left over.
Rewrite as a mixed number:
The whole number is 3 (since 2 fits into 7 three times).
The remainder 1 becomes the numerator of the fraction.
The denominator stays the same (so it remains 2).
So, \(\large \frac{7}{2}\) = \(\large 3\frac{1}{2}\)!
Quite simple, right?
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You might be wondering: If mixed numbers are easier to understand, why would we ever change them back into improper fractions?
Converting from mixed numbers to improper fractions can be helpful when we need to add, subtract, multiply, or divide fractions.
Keeping everything in the form of a fraction makes the math simpler and avoids extra steps with whole numbers.
Let’s go through an example to see how this works.
We’ll take \(\large 2\frac{1}{2}\) and turn it into an improper fraction.
Multiply the whole number (2) by the denominator (2):
2 × 2 = 4
Add the numerator (1) to that product:
4 + 1 = 5
Keep the denominator the same:
The denominator stays 2, so our improper fraction is \(\large \frac{5}{2}\).
So, \(\large 2\frac{1}{2}\) = \(\large \frac{5}{2}\).
That wasn’t so tricky either, right?
Now that we know how to convert improper fractions to mixed numbers and vice versa, the next action is adding mixed numbers.
When adding mixed numbers, we always pay attention to the denominator—the bottom part of the fraction.
If the denominators are the same as in \(\large 2\frac{1}{2} + 3\frac{1}{2}\), we simply:
Add the whole numbers:
2 + 3 = 5
Add the fractions:
\(\large \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1\)
Combine the results:
5 + 1 = 6
\(\large 2\frac{1}{2} + 3\frac{1}{2} = 6\)
Easy peasy!
But when the denominators are different, we need an extra step. Let’s break it down.
We’ll add \(\large 1\frac{3}{4} + 2\frac{1}{2}\) step by step.
Add the whole numbers:
1 + 2 = 3
Find a common denominator for the fractions:
The least common denominator (LCD) of 2 and 4 is 4.
\(\large \frac{3}{4}\) already has a denominator of 4, so we want \(\large \frac{1}{2}\) to have it too.
To do this, we multiply both the numerator and denominator by 2:
\(\large \frac{1}{2} \times \frac{2}{2} = \frac{2}{4}\)
Add the fractions:
Now that both fractions have the same denominator, we add them:
\(\large \frac{3}{4} + \frac{2}{4} = \frac{5}{4}\)
Convert the fraction to a mixed number:
\(\large \frac{5}{4}\) is a mixed number so we divide:
\(\large \frac{5}{4}\) = 1 remainder of 1
1 is the whole number since 4 goes into 5 only 1 time.
The remainder (1) becomes the new numerator, and the denominator stays 4.
We write the mixed number:
\(\large \frac{5}{4} = 1\frac{1}{4}\)
Combine everything:
We add the whole number 3 (from step 1) \(\large 1\frac{1}{4}\):
\(\large 3 + 1\frac{1}{4} = 4\frac{1}{4}\)
So, \(\large 1\frac{3}{4} + 2\frac{1}{2} = 4\frac{1}{4}\).
Subtracting mixed numbers has a pretty similar process to adding mixed numbers.
If the denominators are the same, as in \(\large 5\frac{3}{4} - 2\frac{2}{4}\), we simply:
Subtract the whole numbers:
5 - 2 = 3
Subtract the fractions:
\(\large \frac{3}{4} - \frac{2}{4} = \frac{1}{4}\)
Combine the results:
The final answer is \(\large 3\frac{1}{4}\).
However, when the denominators are different, we need an extra step. Let’s break it down.
We’ll subtract \(\large 4\frac{2}{3} - 1\frac{1}{2}\).
Subtract the whole numbers:
4 -1 = 3
Find a common denominator for the fractions:
The least common denominator (LCD) of 3 and 2 is 6.
To get \(\large \frac{2}{3}\) to have a denominator of 6, we multiply the numerator and the denominator by 2.
\(\large \frac{2}{3} \times \frac{2}{2} = \frac{4}{6}\)
To get \(\large \frac{1}{2}\) to have a denominator of 6, we multiply the numerator and the denominator by 3.
\(\large \frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\)
Subtract the fractions:
Now that both fractions have the same denominator, we subtract:
\(\large \frac{4}{6} - \frac{3}{6} = \frac{1}{6}\)
Combine everything:
The whole number 3 and the fraction 16 give us 316.
So, \(\large 4\frac{2}{3} - 1\frac{1}{2} = 3\frac{1}{6}\)!
Multiplying mixed numbers is a little different from adding and subtracting.
Since mixed numbers contain both whole numbers and fractions, the easiest way to multiply them is to first convert them into improper fractions (just like we did earlier!).
Let’s multiply \(\large 2\frac{1}{2} \times 3\frac{1}{3}\) to show you the whole process:
Convert the mixed numbers to improper fractions:
For \(\large 2\frac{1}{2}\):
Multiply the whole number (2) by the denominator (2): 2 × 2 = 4
Add the numerator (1) to that product: 4 + 1 = 5
Place 5 over the same denominator (2): \(\large 2\frac{1}{2} = \frac{5}{2}\)
For \(\large 3\frac{1}{3}\):
Multiply the whole number (3) by the denominator (3): 3 × 3 = 9
Add the numerator (1): 9 + 1 = 10
Place it over the same denominator (3): \(\large 3\frac{1}{3} = \frac{10}{3}\)
Multiply the fractions:
\(\large \frac{5}{2} \times \frac{10}{3} = \frac{50}{6}\)
Convert the improper fraction to a mixed number:
Divide 50 ÷ 6:
6 goes into 50 a total of 8 times, with a remainder of 2 (8 × 6 = 48).
The whole number is 8, and the remainder 2 becomes the numerator, keeping the denominator 6.
So, \(\large \frac{50}{6} = 8\frac{2}{6}\)
4. Simplify the fraction if needed:
\(\large \frac{2}{6}\) simplifies to \(\large \frac{1}{3}\), so the final answer is \(\large 8\frac{1}{3}\).
Dividing mixed numbers follows the same first step as multiplication—we convert them into improper fractions. Then, instead of dividing, we use the reciprocal (flip the second fraction) and multiply.
To show you the steps in detail, we’ll divide \(\large 2\frac{1}{2} \div 1\frac{1}{3}\).
Convert the mixed numbers to improper fractions:
For \(\large 2\frac{1}{2}\):
Multiply the whole number (2) by the denominator (2): 2 × 2 = 4
Add the numerator (1) to that product: 4 + 1 = 5
Place 5 over the same denominator (2): \(\large 2\frac{1}{2} = \frac{5}{2}\)
For \(\large 1\frac{1}{3}\):
Multiply the whole number (1) by the denominator (3): 1 × 3 = 3
Add the numerator (1): 3 + 1 = 4
Place it over the same denominator: \(\large 1\frac{1}{3} = \frac{4}{3}\)
Flip the second fraction and change the division to multiplication:
Instead of dividing by \(\large \frac{4}{3}\), we multiply by its reciprocal \(\large \frac{3}{4}\).
So the problem becomes: \(\large \frac{5}{2} \times \frac{3}{4}\)
Multiply the fractions:
\(\large \frac{5}{2} \times \frac{3}{4} = \frac{15}{8}\)
Convert the improper fraction to a mixed number:
Divide 15 ÷ 8:
8 goes into 15 once (1 × 8 = 8), with a remainder of 7.
The whole number is 1, and the remainder 7 becomes the numerator, keeping the denominator 8.
So, \(\large \frac{15}{8} = 1\frac{7}{8}\)
Our final result is: \(\large 2\frac{1}{2} \div 1\frac{1}{3} = 1\frac{7}{8}\).
Now it’s your turn! Practice what you’ve learned with these examples.
Convert the improper fraction to a mixed number:
\(\large \frac{11}{3}\) = ?
Convert the mixed number to an improper fraction:
\(\large 4\frac{2}{5}\) = ?
Add the mixed numbers:
\(\large 2\frac{3}{5} + 1\frac{3}{5}\) = ?
Subtract the mixed numbers:
\(\large 5\frac{3}{4} - 2\frac{1}{2}\) = ?
Multiply the mixed numbers:
\(\large 1\frac{2}{3} \times 2\frac{1}{2}\) = ?
Divide the mixed numbers:
\(\large 3\frac{2}{5} \div 1\frac{1}{5}\) = ?
When you’re done, check your answers at the bottom of the guide
Learning about mixed numbers doesn’t come without a few dilemmas! As students explore how to convert, add, subtract, multiply, and divide mixed numbers, here are a few questions we usually get at Mathnasium of Mason:
No, because then it would be an improper fraction!
A mixed number always has a proper fraction (where the numerator is smaller than the denominator) next to the whole number.
For example:
\(\large 3\frac{2}{5}\) is a mixed number because 2 < 5.
\(\large 3\frac{6}{5}\) is incorrect as a mixed number because 6 > 5—it should be written as \(\large 4\frac{1}{5}\) instead.
Yes! Just like with whole numbers, subtracting a larger number from a smaller one gives a negative result.
For example, let’s subtract \(\large 4\frac{1}{3} - 6\frac{1}{3}\):
Subtract the whole numbers:
4 - 6= -2
Subtract the fractions:
\(\large \frac{1}{3} - \frac{1}{3} = 0\)
Combine everything:
Since the fractions cancel each other out, the final answer is -2.
So, \(\large 4\frac{1}{3} - 6\frac{1}{3} = -2\).
Not exactly! A mixed number shows a whole number and a fraction, while a decimal represents a number in base ten.
For example:
\(\large 2\frac{1}{5}\) is a mixed number.
2.20 is its decimal equivalent.
Even though they represent the same quantity, they are written in different forms.
Mixed numbers are often used in measurements (like recipes and lengths), while decimals are common in money and calculations.
Mathnasium of Mason is a math-only learning center for K-12 students of all skill levels in Mason, OH.
Using a proprietary teaching approach called the Mathnasium Method™, our specially trained math tutors offer face-to-face instruction in an engaging and fun environment to help students master any math topic, including mixed numbers—typically introduced in the 4th grade.
Students begin their Mathnasium journey with a diagnostic assessment that allows us to understand their unique strengths and knowledge gaps. Guided by assessment-based insights, we create personalized learning plans that will put them on the best path toward math mastery.
Whether your student is looking to catch up, keep up, or get ahead in their math class, schedule an assessment, and enroll at Mathnasium of Mason today!
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If you’ve given our exercises a try, find your answers below:
\(\large \frac{11}{3} = 3\frac{2}{3}\)
\(\large 4\frac{2}{5} = \frac{22}{5}\)
\(\large 2\frac{3}{5} + 1\frac{3}{5} = 4\frac{1}{5}\)
\(\large 5\frac{3}{4} - 2\frac{1}{2} = 3\frac{1}{4}\)
\(\large 1\frac{2}{3} \times 2\frac{1}{2} = 4\frac{1}{6}\)
\(\large 3\frac{2}{5} \div 1\frac{1}{5} = 2\frac{5}{6}\)
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