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High School Algebra I: Help and Review25 chapters | 288 lessons
Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.
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're given. One way is to multiply the numerator and denominator by the same number. For example, the fraction 1/2 is equivalent to 3/6 because if I multiply both 1 and 2 by 3, I get 3 in the numerator and 6 in the denominator.
(1*3)/(2*3) = 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 two 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. 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 I now 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 reduce to the fraction 3/2. In fact, all equivalent fractions will always reduce to the same fraction in lowest terms.
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's better to use the GCD because if you don't 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 be 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:
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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