Comparing Mixed Numbers

Do you remember when you had to compare fractions? You're going to get problems asking you to compare mixed numbers and tell which is bigger or smaller. As always, sometimes it's really easy. The first step is to look at the whole numbers. If the whole numbers are different, just compare those values.

Examples:
Compare 5 1/2 and 17 1/2.
They may both have one-half as the fraction, but it doesn't matter. 17 is greater than 5.
5 1/2 < 17 1/2

Compare 3 99/100 and 12 6/100.
Again, it doesn't matter about the fractions. It does not matter if 99/100 is greater than 6/100. You only need to worry about the whole numbers right now. 3 is less than 12.
3 99/100 < 12 6/100


If the whole numbers are the same, you only need to worry about the fractions. In the next example, both whole numbers are 2, so we can ignore them. You will only need to compare the fractions once you have created common denominators.

Example:
Compare 2 3/8 and 2 5/11.
Since the whole numbers are both 2, that doesn't help us at all. When we focus on the fractions, we quickly see that they are unlike with denominators of 8 and 11. They don't even have common factors. We'll have to multiply each fraction with the denominator of the other. Do you remember how we multiply with equivalents of 1 to make common denominators?

3/8 = 3/8 * 1 = 3/8 * 11/11 = (3*11)/(8*11) = 33/88

5/11 = 5/11 * 1 = 5/11 * 8/8 = (5*8)/(11*8) = 40/88

We know that 33/88 < 40/88, so 3/8 < 5/11. Since the whole numbers are the same, we also know...
2 3/8 < 2 5/11

Adding and Subtracting

Adding and subtracting mixed numbers is pretty easy. There are a lot of methods you can use. Some create improper fractions at the beginning of the process and some do it at the end. Here's an example where you start with improper fractions and then simplify your answer.

Example:
2 5/7 + 3 6/7 = ?
Steps to Solve:
• Make improper fractions: 2 5/7 can be converted into 19/7, and 3 6/7 converts to 27/7.
• Do the fractions have common denominators? Yes. Nothing to do here.
• Add the numerators: 19 + 27 = 46. Your answer is the improper fraction 46/7.
• Convert the improper fraction to a mixed number by doing division and having a remainder. 46√∑7 = 6r4
• Write the quotient (answer to the division problem) as a mixed number.
Answer:
6 4/7


You will use the same process for subtraction. You'll just be subtracting the numerators instead of adding them.

Multiplying and Dividing

Multiplying and dividing mixed numbers use the same steps as multiplying and dividing regular fractions, but you don't have to worry about common denominators. Just convert the mixed numbers into improper fractions and get to the fraction arithmetic. We'll have more examples in the next sections.

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