- Topics At a Glance
- Different Types of Numbers
- Natural Numbers
- Whole Numbers
- Integers and Negative Numbers
- Integers and Absolute Value
- Rational Numbers
- Irrational Numbers
- Real Numbers and Imaginary Numbers
**Different Ways to Represent Numbers**- Fractions
- Equivalent Fractions
- Mixed Numbers
**Reducing Fractions**- Comparing Fractions
- Least Common Denominator
- Addition and Subtraction of Fractions
- Multiplication of Fractions
- Equivalent Fractions and Multiplication by 1
- Multiplication by Clever Form of 1
- Multiplicative Inverses
- Division of Fractions
- Multiplication and Division with Mixed Numbers
- Decimals
- Converting Fractions into Decimals
- Converting Decimals into Fractions
- Comparing Decimals
- Adding and Subtracting Decimals
- Multiplication and Division by Powers of 10
- Multiplying Decimals
- Dividing Decimals
- Infinite Decimals
- Percents
- Portion of the Whole
- Things to Do with Real Numbers
- Addition and Subtraction of Real Numbers
- Properties of Addition
- Subtraction
- Multiplication
- Division
- Long Division Remainder
- Exponents and Powers - Whole Numbers
- Properties of Exponents
- Prime Factorization
- Order of Operations
- Even and Odd Numbers
- Infinity
- Sequences
- is Irrational
- Counting Rational Numbers
- Counting Real Numbers?
- Counting Irrational Numbers
- In the Real World
- Decimals in Use
- How to Solve a Math Problem
- I Like Abstract Things: Summary

There are many ways to represent any one fraction. How do we choose which to use? It will depend on the problem, but it's often helpful to **simplify **or **reduce** the fraction.

We know that if we start with a fraction and multiply the numerator and denominator each by the same number *n*, we get a fraction equivalent to . This also works in reverse: if the whole number *n* divides evenly into both *p* and *q*, then dividing *p* and *q* each by *n* will create a new fraction equivalent to . Let's call it *j/b*, for a change of pace.

Consider the fraction .

4 and 6 are both divisible by 2. Divide each by 2 to get a fraction equivalent to .

Easy peasy - that's .

To simplify or reduce a fraction , we carry out this process until there are no natural numbers *n* left (except 1) that divide both *p* and *q*. It is now in **lowest terms. **This fraction would win the limbo every time.

How to systematically reduce fractions:

Rather than just eyeballing a couple of numbers and using the trial and error method, we can save ourselves some time by finding the biggest number *n* that divides both the numerator and the denominator. Ho there - this is something we've seen before! The number *n* is the **GCF [link to Pre-algebra section about GCF]**, or the greatest common factor of the numerator and denominator! In the improv world, we call this a "callback." (We like to pretend we're a part of the improv world. Just humor us.)

To quickly figure out what the GCF of the numerator and denominator of a fraction is, write out the **prime factorizations** (link) of the numerator and denominator. The GCF is what you get when you multiply all the prime factors shared by both numerator and denominator. It's all about the overlap.

We can see that 3 divides both the numerator and the denominator. When we go ahead and divide the numerator and denominator each by 3 we get . This process is also called "canceling out" the 3's from the top and bottom of the fraction. Apparently, they didn't get very good ratings. They were really just appealing to the prime number demographic.

Check the overlap - we can cancel out one 2 and one 3 from both the numerator and denominator to get

There is a huge benefit to this process of canceling out the common prime factors to arrive at a reduced fraction. Since a number is completely determined by its prime factorization, the upshot is that, even though there are many, many equivalent fractions representing the same rational number, there is only *one* fully reduced fraction! It's like you've been dating around and have met plenty of perfectly nice guys, but finally you've found "the one!" And all you had to do was reduce him to practically nothing! Oh happy day!

Example 1

Reduce to its lowest terms. |

Example 2

Reduce to its lowest terms. |

Exercise 1

Reduce .

Exercise 2

Reduce .

Exercise 3

Is in lowest terms?

Exercise 4

Is in lowest terms?

Exercise 5

Is in lowest terms?

Exercise 6

Is in lowest terms?

Exercise 7

Simplify .

Exercise 8

Simplify .

Exercise 9

Simplify .

Exercise 10

Simplify .

Exercise 11

Simplify .