Simplifying Fractions

What is the Simplified Form of a Fraction?

We're all equal here!

Any rational number (meaning a fraction) can be written in an infinite number of ways. For instance, one half is the same as 2 quarters, 3 sixths, 4 eighths, and so on. The simplified form is when you write it with the smallest possible numbers. In our example, the simplest form is one half. Although a fraction has an infinite number of equivalent forms, it only has one simplest form.

How do you Simplify Fractions?

There’s a variety of ways to simplify fractions; in the end, they all boil down to the same thing.

Simplifying a Fraction by Prime Factorization:

  1. Find the prime factorization of the numerator.
  2. Find the prime factorization of the denominator.
  3. Write a big fraction with all the numerator’s factors on top, and all the denominator’s factors on the bottom. Separate the factors with multiplication symbols to keep your work clean and avoid mistakes. If there are multiple copies of factors it’s OK to save time and space by using exponents.
  4. If any factor is the same on the top and the bottom, then cross it off in both places. If there are multiple copies of some factor, you can only cross off as many in one place as there are in the other. For example, if there’s three 2s on the top, and four 2s on the bottom, then all three 2s on the top can be crossed off, but only cross off three of them on the bottom.
  5. Repeat step 4 until there are no factors shared between the top and the bottom.
  6. Multiply all the remaining factors on the top, and write that as the numerator of the answer.
  7. Multiply all the remaining factors on the bottom, and write that as the denominator of the answer.
  8. If all the factors were crossed off in the top or the bottom, then what is left there is the number one.

Let's simplify! Find the prime factorization, take away common factors, then multiply, and now it's simplified!

Simplifying a Fraction One Factor at a Time:

  1. Look at the numerator and denominator trying to see any factors that are common to both.
  2. Divide both top and bottom by the common factor; write down the new fraction.
  3. Repeat steps 1 and 2 until there are no longer any common factors.
  4. Make a clean copy of the final answer. (Sometimes when this technique is used, it can get messy with many crossed out temporary steps.)

Let's simplify! Take away a number that is in both the top and bottom, and then do it again, and again, and again, and now it's simplified!

Which technique to use?

In simple cases, one tries to see the GCF of both numbers and complete the simplification in a single step. On the other hand, when solving a problem in which seeing common factors is hard, the other technique, simplifying by prime factorization, might work better.

Why do we Simplify Fractions?

There’s more than one reason why simplifying fractions is a good idea.

  • Simplified fractions use smaller numbers. Smaller numbers means that if you do more arithmetic with the fraction later, life will be easier, and you will have less work to do.
  • It can be hard to tell if two equivalent fractions are actually the same if they are written in different forms. Since each fraction only has one simplest form, you can tell if simplified fractions are the same just by looking at them.
  • Simplified fractions are definitely the form you want for the answers to real life problems. For example, if you were figuring out the length to cut a piece of wood for a carpentry project, or how much flour to use for a baking project you would most certainly want to use the simplified form for your measurements.
  • Sometimes a school math assignment is merely to simplify fractions without doing anything else with them. Of course, even though nobody says the reason, the real goal in such cases is to get familiar with working with fractions, practice factoring, and memorize the steps. A lot goes into being able to simplify fractions! Practicing the simplification of fractions as separate problems at first makes more complicated problems, that involve simplification as just one of many steps, less confusing.
  • Recipe for Chocolate Chip Cookies. The quantities use fractions that need to be simplified.

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