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*In this lesson, we will learn about working with improper fractions. Although a topic of elementary mathematics, fractions are important and occur in everyday calculations. Knowing how to calculate with fractions is an important lifelong skill.*

We also recommend watching What is Simplest Form? - Definition & How to Write Fractions in Simplest Form and

All fractions contain two parts, a numerator and denominator. In the fraction 2/3, the number '2' is the numerator and the number '3' is the denominator. The denominator tells how many equal-sized parts make one whole and the numerator tells how many of those parts are being counted. So, 2/3 means that a whole contains 3 equal sized parts and only 2 of the 3 are being counted.

Sometimes, the number of parts being counted is actually more than the number of parts in the whole. The fraction 4/3 is an example of this and these types of fractions are called **improper fractions**. Since you only need 3 parts to make 1 whole and 4 parts are being counted, the fraction 4/3 actually represents a number greater than 1 whole, and this is true for all improper fractions.

You can make many fractions that are **equivalent**, or equal in value, to one that you are given. One way is to multiply the numerator and denominator by the same number. For example, the fraction ½ is equivalent to 3/6 because if I multiply both 1and 2 by 3, I get 3 in the numerator and 6 in the denominator.

- (1x3)/(2x3) = 3/6

These fractions represent the same value. I can also make an equivalent fraction by dividing the numerator and denominator by the same number. For example, consider the fraction 6/4. I can divide both the numerator and denominator by the number 2.

- (6/2)/(4/2) = 3/2

This means that the fraction 3/2 is equivalent or equal to the fraction 6/4.

To demonstrate that these two fractions represent equal amounts, two diagrams are shown. In one, we see 6/4 represented as 2 rectangles, where each is divided into 4 equal parts and 6 parts total shaded. The second diagram shows two equal sized rectangles but divided differently--each rectangle is divided into 2 equal sized parts and 3 parts total shaded. Although they are divided differently, the two rectangles can be seen to represent the same amount.

The terms 'simplifying' or 'reducing' means the same thing when referring to fractions, so these terms can be used interchangeably.

When you reduce a fraction, it becomes simpler because the number of parts in the whole is made small as possible, without changing the value of the fraction. Returning to the previous example, we found that 6/4 = 3/2. While both represent the same amount, 3/2 has 2 parts to make one whole and 2 is smaller than the 4 parts to make one whole in the fraction 6/4.

It is possible to make a fraction simpler without completely simplifying it. Consider, for example, the fraction 18/12. To make a simpler fraction, I could divide the top and bottom by 3.

- (18/3)/(12/3) = 6/4

This fraction is simpler because now I have 4 parts to make one whole instead of 12. However, we know from working with this same fraction above that it can be simplified further to 3/2.

A fraction is **simplified** or **reduced to lowest terms** when there is no number other than 1 that divides into both the numerator and denominator. We would say that 6/4 and 18/12 reduces to the fraction 3/2. In fact, all equivalent fractions will always reduce to the same fraction in lowest terms.

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To reduce a fraction to lowest terms, try to find the largest number that divides into both numerator and denominator. This number is also known as the greatest common divisor or GCD for both the numerator and denominator. The multiplication tables may be helpful. Notice, for example, that both 18 and 12 are in the 6 time tables and it is the largest multiplication table that contains both 18 and 12. So, to reduce this fraction we divide both 18 and 12 by 6.

- (18/6)/(12/6) = 3/2

It is better to use the GCD because if you do not use the GCD first, more than one division step will be necessary. After dividing once, check again for another number that can be divided into numerator and denominator.

Remember, the numerator and denominator must be divided by the same number or the new fraction will not equivalent to the original one.

A fraction can also be reduced by factoring the numerator and denominator into prime factors, then cancelling all common factors.

To show this method on 18/12, the steps would be as follows:

- Factor 18 into prime factors. 18 = 3x3x2
- Factor 12 into prime factors. 12 = 3x2x2
- Rewrite the numerator and denominator as a product of prime factors. 18/12 = (3x3x2)/(3x2x2)
- Cancel all factors common to the numerator and denominator. The remaining factors will be the reduced fraction. In this case, one factor of 3 and 2 are common and can be cancelled. (3x2x3)/(3x2x2) = 3/2

An improper fraction is one where the numerator is larger than the denominator. All improper fractions are greater than one whole. To reduce an improper fraction, you must use division to find an equivalent fraction with the smallest number of parts in the whole. One method is to divide the numerator and denominator by the same number. They must be divided by the same number or the new fraction will not equivalent to the original one. A fraction can also be reduced by factoring both numerator and denominator into prime factors, then cancelling all common factors.

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