How To Reduce Mixed Numbers & Improper Fractions To The Lowest Terms

When you see the term "improper fraction," it doesn't have anything to do with etiquette. Instead, it means that the numerator, or top number, of the fraction is larger than the denominator, or bottom number. Depending on the instructions for the problem you're working on, you can keep an improper fraction in that form, or you can convert it to a mixed number: A whole number paired with a proper fraction. Either way, your math life will be a lot easier if you get into the habit of reducing all of those fractions to lowest terms.

Should you keep improper fractions the way they are, or convert them into a mixed number? That depends on the instructions you get and your ultimate goal. As a general rule, if you're still doing arithmetic with the fraction, it's easier to leave it in improper form. But if you're done with the arithmetic and ready to interpret your answer, it's easier to convert the improper fraction to a mixed number by working the division it represents.

1. Work the Division

Recall that you can also write a fraction as division. For example, 33/12 is the same as 33 ÷ 12. Work the division the fraction represents, leaving your answer in remainder form. To continue with the example given:

\(33 ÷ 12 = 2 \text{, remainder } 9\)

2. Write the Remainder as a Fraction

Write the remainder as a fraction, using the same denominator as your original fraction:

\(\text{remainder } 9 = \frac{9}{12}\)

because 12 was the original denominator

3. Combine a Whole Number and Fraction

Finish writing the mixed number as a combination of the whole number result from Step 1, and the fraction from Step 2:

\(2 \enspace \frac{9}{12}\)

Whether you're dealing with improper fractions or the fraction part of a mixed number, simplifying the fraction to lowest terms makes them easier to read and easier to work arithmetic with. Consider the fraction part of the mixed number you just calculated

\(\frac{9}{12}\)

1. Look for Common Factors

Look for factors that are present in both the numerator and the denominator of the fraction. You can either do this by examination (looking at the numbers and listing their factors in your head) or by writing out the factors for each number. Here's how you'd write the factors out:

Factors of 9: 1, 3, 9

Factors of 12: 1, 3, 4, 12

2. Find the Greatest Common Factor

Whether you're using examination or a list, find the greatest factor that both numbers share. In this case, the greatest factor present in both numbers is 3.

3. Divide by the Greatest Common Factor

Divide both numerator and denominator by the greatest common factor or, to think of it another way, factor that number out of both numerator and denominator and then cancel it. Either way, you end up with:

\(\frac{9 ÷ 3}{12 ÷ 3} = \frac{3}{4}\)

Because the numerator and denominator no longer have any common factors greater than 1, your fraction is now in lowest terms.

The process works exactly the same for simplifying an improper fraction to lowest terms. Consider the improper fraction

\(\frac{25}{10}\)

1. Look for Common Factors

Examine both numbers, or make a list, to find their factors:

Factors of 25: 1, 5, 25

Factors of 10: 1, 2, 5, 10

2. Find the Greatest Common Factor

In this case, the greatest factor that's in both numbers is 5.

3. Divide by the Greatest Common Factor

Divide both numerator and denominator by 5. This gives you:

\(\frac{5}{2}\)

Because 5 and 2 share no common factors greater than 1, the fraction is now in lowest terms.