Find practice problems and solutions for these concepts at Operations with Mixed Numbers Practice Problems.
What's Around The Bend
 What's a Mixed Number?
 Converting Improper Fractions to Mixed Numbers
 Converting Mixed Numbers to Improper Fractions
 Adding Mixed Numbers
 Subtracting Mixed Numbers
 Multiplying Mixed Numbers
 Reciprocals of Mixed Numbers
 Dividing Mixed Numbers
Part Whole Number, Part Fraction
So far, we've looked at proper fractions—fractions that are between –1 and 1—and improper fractions—fractions that are greater than or equal to 1, or less than or equal to –1. We can rewrite improper fractions as mixed numbers.
Improper Fractions Become Mixed Numbers …
Remember, the fraction bar represents division. means "5 divided by 8." To convert an improper fraction to a mixed number, divide the numerator by the denominator.
Example
In this improper fraction, the numerator is 7 and the denominator is 3; 7 divided by 3 is 2, with 1 left over. The whole number part of our answer is 2, and we express the remainder as a fraction. Because the improper fraction has a denominator of 3, our remainder has a denominator of 3; 7 divided by 3 is .
Example
Divide the numerator by the denominator: 11 divided by 4 is 2, with 3 left over, which means that our mixed number is .
Sometimes, an improper fraction can be converted to a whole number. After dividing the numerator by the denominator, there is no remainder, so there is no fractional part of our answer, just a whole number.
Example
15 divided by 5 is 3, with no remainder. Our answer is a whole number, not a mixed number.
… And Mixed Numbers Become Improper Fractions
Just as we can convert improper fractions to mixed numbers, we can convert mixed numbers to improper fractions. This will soon become important, as we learn to add, subtract, multiply, and divide mixed numbers.
There are three steps to converting a mixed number to an improper fraction. First, multiply the whole number by the denominator of the fraction. Second, add to that product the numerator of the fraction. Third, put that sum over the denominator of the fraction.
Example
We begin by multiplying the whole number, 5, by the denominator of the fraction, 4: 5 × 4 = 20. Next, add to that product the numerator of the fraction: 20 + 3 = 23. Finally, put that sum over the denominator of the fraction:
Example
Follow the three steps. First, 8 × 9 = 72. Next, 72 + 2 = 74.
Finally, .
Adding Mixed Numbers
Now let's look at adding mixed numbers. Knowing how to convert improper fractions to mixed numbers will help us simplify our answers.
To add two mixed numbers, first, add the whole number parts of each number. Then, add the fractions. If the fractions are unlike, we may need to find common denominators. Finally, if the sum of the fractions is an improper fraction, convert that fraction to a mixed number and add it to the whole number part of your answer.
Example
First, add the whole number parts of each number: 4 + 6 = 10. Next, add the fractions. Because these fractions are like, just add the numerators and keep the denominator. . This sum is a proper fraction, so our answer is .
Here's an example where the sum of the fractions is an improper fraction—we'll have a little more work to do!
Example
Again, begin by adding the whole number parts of each number: 3 + 1 = 4. Next, add the fractions: . Now, convert to a mixed number; 6 divided by 4 is 1, with 2 left over: , which reduces to . Finally, add this mixed number to the whole number part of our answer by adding the whole number parts, and keeping the fraction:
.
The denominators of the fractions in mixed numbers can be unlike, just as when we looked at adding fractions in the last chapter. And just as in the last chapter, to add mixed numbers with unlike denominators, we must find common denominators before adding.
Example
First, add the whole numbers: 9 + 15 = 24. Next, add the fractions. The least common denominator of 3 and 15 is 15: . Now, we can add the fractions: . Because 17 divided by 15 is 1 with 2 left over, . Finally, add this mixed number to the sum of the whole numbers: .
Subtracting Mixed Numbers
We can use a similar strategy to subtract mixed numbers. In a subtraction problem involving two mixed numbers, subtract the whole number part of the second mixed number from the whole number part of the first mixed number. Then, subtract the fraction part of the second mixed number from the fraction part of the first mixed number.
Example
Work with the whole numbers first: 9 – 2 = 7. Next, work with the fractions. . The difference is a simplified, proper fraction, so our answer is .
We've seen how the sum of two mixed numbers can be just a whole number. The same goes for differences.
Example
Work with the whole numbers first: 5 – 3 = 2. Next, work with the fractions. = 0. Our answer is simply the whole number 2.
As you might have guessed, the difference between two mixed numbers might not contain a whole number: is equal to , because 4 – 4 = 0.
Just as you need common denominators to add mixed numbers, you need common denominators to subtract mixed numbers.
Example
Work with the whole numbers first: 10 – 2 = 8. Next, work with the fractions. The least common multiple of 10 and 4 is 40. and . Now we can subtract: . The greatest common factor of 6 and 40 is 2, so reduces to . Our answer is .
Subtracting mixed numbers can get a little tricky when the fraction part of the mixed number that is being subtracted is greater than the fraction part of the mixed number from which it is being subtracted. When that happens, we must borrow from the whole number part of our answer.
Example
Work with the whole numbers first: 5 – 1 = 4. Now look at the fractions. We can't subtract from , because is greater than . We need to borrow 1 from the whole number part of our answer: 4 – 1 = 3. The whole number part of our answer is now 3. What do we do with the 1 we just borrowed? We write it as and add it to the fraction that we already have, . Now we can subtract the fractions: . The fractional part of our answer is . Therefore, .
Instead of subtracting from , we subtracted from , because and are equal. When we borrow 1 from the whole number part of the mixed number, we write that 1 as a number over the denominator of the mixed number. Because the denominator of the mixed number was 3, we wrote the borrowed 1 as .
Let's look at another example just to be sure we've got it down!
Example
Again, work with the whole numbers first: 15 – 12 = 3. Now look at the fractions. The least common denominator of 7 and 8 is 56; and . We can't subtract from , because is greater than . We need to borrow 1 from the whole number part of our answer: 3 – 1 = 2. We write the borrowed 1 as and add it to the fraction that we already have: . Now we can subtract the fractions: . The fractional part of our answer is . Therefore, .
Some people prefer to convert both mixed numbers to improper fractions before subtracting. If you do this, you won't have to worry about borrowing at all. Converting mixed numbers to improper fractions is good practice, and as we'll soon see, it's the only way to multiply and divide mixed numbers. Let's look at a subtraction example first.
Example
First, convert each number to an improper fraction. 3 × 2 = 6, 6 + 1 = 7, so . 2 × 5 = 10, 10 + 4 = 14, so . The least common denominator of 2 and 5 is 10; and . Now we can subtract: .
Inside Track
If you can see that a subtraction problem involving mixed numbers will NOT require borrowing, the easiest method is to subtract whole numbers and subtract fractions. If you can see that the problem will require borrowing, the easiest method is to convert each mixed number to an improper fraction before subtracting. However, either method will work in either case. If you find one method easier than the other, use it all the time!

Multiplying Mixed Numbers
There is only one method for multiplying mixed numbers: Convert each mixed number to an improper fraction, and then multiply the numerators and multiply the denominators. No common denominators needed—once we have two improper fractions, we're ready to go!
Example
Convert each number to an improper fraction. 2 × 6 = 12, 12 + 1 = 13, so ; 6 × 3 = 18, 18 + 2 = 20, so . Now that we have two improper fractions, multiply the numerators and multiply the denominators: 13 × 20 = 260 and 6 × 3 = 18, so . The greatest common factor of 260 and 18 is 2, so reduces to . 130 divided by 9 is 14 with 4 left over, so .
Let's look at one more example.
Example
Convert each number to an improper fraction: 8 × 7 = 56, and 56 + 4 = 60, so ; 3 ×5 = 15, and 15 + 3 = 18, so . Now we have the multiplication problem . Divide the 60 in the first fraction and the 5 in the second fraction by 5. The problem becomes × 18; 12 × 18 = 216; 216 divided by 7 is 30 with 6 left over, so .
Pace Yourself
A pizzeria needs to cater 6 different parties, and each party needs pizzas. How many pizzas does the pizzeria need to make? What if the pizzeria was catering 9 parties? What if each party needed pizzas? Besides this example of a pizzeria, what other reallife situations can you think of that would require multiplying mixed numbers?

Dividing Mixed Numbers
You've probably noticed the similarities between adding, subtracting, multiplying, and dividing fractions and adding, subtracting, multiplying, and dividing mixed numbers. You might have already guessed how to divide mixed numbers: Convert them to improper fractions, find the reciprocal of the divisor, and multiply. No common denominators (and no division!) needed.
Caution!
Because Mixed Numbers must be converted to improper fractions before dividing, it would be easy to forget to take the reciprocal of the second fraction (the divisor) before dividing. Don't let that happen to you! Start a division problem with mixed numbers by converting the divisor to an improper fraction FIRST—and then find its reciprocal right away. Then, convert the first fraction (the dividend) to an improper fraction. Now you're ready to multiply.

Example
Let's use the tip we just learned: convert the divisor, , to an improper fraction, and then find its reciprocal. 4 × 10 = 40, 40 + 7 = 47, so . Remember, to find the reciprocal of a fraction, switch the numerator and the denominator. The reciprocal of is . Now convert the dividend, , to an improper fraction. 1 × 8 = 8, and 8 + 6 = 14, so . Our problem has become . Divide the 8 in the first fraction and the 10 in the second fraction by 2, and the problem becomes . We can also divide the 14 in the first fraction and the 4 in the first fraction by 2, making the problem . Multiply the numerators and multiply the denominators: 7 × 5 = 35 and 2 × 47 = 94, making our answer .
By first learning how to add, subtract, multiply, and divide fractions, we established a solid foundation for learning how to add, subtract, multiply, and divide mixed numbers. After all, performing those operations on mixed numbers was just like working with fractions—just really large, improper fractions.
Find practice problems and solutions for these concepts at Operations with Mixed Numbers Practice Problems.
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