Reducing Fractions

Reducing fractions is an essential part of reaching the final solution in any fraction problem. However, the process can seem intimidating and not merely because it’s fractions. Reducing fractions refers to the same thing as simplifying fractions.

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Why Reducing Fractions Matters

When people solve fractions, they end up with a wide variety of denominators. It’s a natural part of the fraction problem-solving process. Unfortunately, that makes grading difficult, since each problem would have an infinite number of potential answers.

Instead, reducing the fractions means everyone’s final answer is universal if they did the problem correctly. It also makes it easier to talk about fractions, since reduced fractions have the smallest numbers possible as well.

A Review of Fraction Basics

Before we jump into reducing fractions, it’s critical you understand the terminology for fractions. Otherwise, you may get frustrated before you jump into fraction operations. The top of a fraction is called a numerator, while the bottom is called a denominator. Many people think of it as denominator down, which associates the ds.

Also, for reference in the future, the bar in a fraction is a division sign. That means every fraction problem is a division problem. When talking about algebra or converting fractions to decimals, this becomes more important.

Fraction Structure Diagram

The Steps for Reducing Fractions

The first step involves solving the problem. Remember to be careful about which operation the question asks you for and always double-check your math for errors before beginning.

Now that you have a fraction, it’s time to get to work. You’ll want to write down the fraction clearly so you can look at it. If necessary, lay your multiplication table next to your work surface. Now you’re going to look for numbers on your table that match those in your fraction.

The trick is finding numbers that use the same factors since you must divide the top and the bottom of the fraction by the same number. A factor is a number you multiply against another number.

You may need to repeat the dividing step a few times in order to get a fraction to its most reduced form. There is nothing wrong with needing multiple steps to accomplish the task. You will know you are done reducing when you can no longer find a common factor.

Examples

Let’s start with some examples, which should clear up how reducing fractions works.

Let’s start with a problem where the answer is 15/30. The possible factor combinations for 15 are 1 times 15 and 3 times 5. Meanwhile, there are more possible factors for 30. These include 1 times 30, 2 times 15, 3 times 10, and 5 times 6. The greatest factor the sets have in common is 15, so let’s divide by that. Dividing both the bottom and the top by turns the 15/30 into 1/2. The number 1 is prime, and there’s no point in dividing by 1. That makes 1/2 the final reduced form.

Reducing Fractions Example 1

Now for a slightly more complicated problem, 8/48. The factors for 8 are straightforward at 1 times 8 or 2 times 4. The numbers for 48 are more complicated, with 1 times 48, 2 times 24, 3 times 16, 4 times 12, and 6 times 8. Since 8 is the largest factor, let’s divide by that. The 8/48 becomes 1/6. Since 1 is prime and 6 doesn’t go in it, the final answer is 1/6.

Reducing Fractions Example 2

Then there are the complicated ones, like 192/345. Generally, you don’t want to list all the factors out for big numbers. Instead, use the rules described below. In this case, 1+9+2=12 and 3+4+5=12, so both the top and bottom are divisible by three. That brings us to 64/115 after division. After checking with all the rules, 64/115 cannot be reduced further and is the final answer.

Reducing Fractions example 3

Divisibility Rules

For the purpose of this list, divisible means evenly divisible.

If it ends in 2, 4, 6, 8, or 0 (even numbers), it is divisible by two.

If the sum of the digits is divisible by 3, then the number is divisible by three. For example, the figures of 123 go 1+2+3=6, which is divisible by three. Therefore 123 is divisible by three.

If the final two digits of a number are divisible by 4, the whole number is divisible by 4. For example, in 3424, the 24 is divisible by 4; therefore, that whole number is.

If the number ends in 5 OR 0, the number is divisible by five.

If a number is divisible by both 2 AND 3, it is divisible by six.

If the last three digits in a number are divisible by 8, the whole number is divisible by eight.

If the number ends in 0, it is divisible by ten.