What do the fractions 1/2, 2/4, 3/6, 150/300 and 248/496 have in common? They're all equivalent, because if you reduce them all to their simplest form, they all equal the same thing: 1/2. In this example, you'd simply factor out the greatest common factors from both numerator and denominator until you arrived at 1/2. But there are other ways in which a fraction can become complicated. No matter what's keeping your fraction from existing in its simplest form, the solution is to remember that you can perform almost any operation on a fraction, as long as you do the same thing to both the numerator and the denominator.
Removing Common Factors
The most common reason you'll be asked to write a fraction in its simplest form is if both the numerator and the denominator share common factors.
List the Common Factors
Write out the factors for the numerator of your fraction, then write out the factors for the denominator. For example, if your fraction is 14/20, the factors for numerator and denominator are are:
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14: 1, 2, 7, 14
20: 1, 2, 4, 5, 10, 20
Identify the Largest Common Factor
Identify any common factors greater than 1. In this example, the largest factor that both numbers have in common is 2.
Divide by the Largest Common Factor
Divide both numerator and denominator of the fraction by the largest common factor. To continue the example, 14 ÷ 2 = 7 and 20 ÷ 2 = 10, so your new fraction becomes 7/10.
Because you performed the same operation on both the numerator and the denominator of the fraction, it's still equivalent to the original fraction. Its value hasn't changed; only the way you write it has changed.
Check for Other Common Factors
Check your work to make sure you're done. If the numerator and denominator don't share any common factors greater than one, the fraction is in its simplest form.
Simplifying Fractions With Radicals
There are a few other "complications" that are very common when you first start dealing with fractions. One is when a radical or square root sign shows up in the denominator of the fraction:
In this case, a could stand for any number; it's just a placeholder. And no matter what that number underneath the radical sign is, you use the same procedure to remove the radical from the denominator, which is also known as rationalizing the denominator. You multiply the denominator by the same radical it already contains, taking advantage of the property that √a × √a = a, or to put it another way, when you multiply a square root by itself you effectively erase the radical sign, leaving yourself with just the number (or in this case, the letter) underneath.
Of course you can't perform any operation on the denominator of the fraction without also applying the same operation to the numerator, so you have to multiply both top and bottom of the fraction by √a. This gives you:
2_√a_/(√a × √a) or, once you've simplified it, 2_√a_/a.
In this case you can't get rid of the square root entirely, but at this stage of math, radicals are usually okay in the numerator but not the denominator.
Simplifying Complex Fractions
Another common obstacle you might encounter to writing a fraction in its simplest form is a complex fraction – that is, a fraction that has another fraction in either its numerator or its denominator, or both. In this case, it helps to remember that any fraction a/b can also be written as a ÷ b. So instead of getting confused if you see something like 1/2 / 3/4, you can start by writing it out with the division sign:
1/2 ÷ 3/4
Next, remember that dividing by a fraction is the same as multiplying by its inverse. Or, to put it another way, you'll get the same result if you flip that second fraction upside down (creating the inverse) and multiply by that, which is a much easier operation to perform. So your operation becomes:
1/2 × 4/3 = 4/6
Note that you're back to a simple fraction – there are no "extra" fractions hiding in the numerator or denominator – but it's not quite in lowest terms. You can also factor 2 out of both numerator and denominator, which gives you 2/3 as your final answer.