How to Multiply and Divide Any Three or more Positive and Negative Fractions

In many Math Tests several problems involve the addition, subtraction, the MULTIPLICATION and the DIVISION of fractions. Most if not all these Math Tests are timed. This Article will show step-by-step how to MULTIPLY or DIVIDE any three or more positive and negative fractions QUICKLY.

Things You'll Need

• Pencil and
• Paper

Instructions

1. Given any three or more Fractions ( A / B ), ( C / D ), and ( E / F ) where A, B, C, D, E, and F are Integers. Integers are the Set of negative and positive whole numbers including zero ( 0 ). The denominators of the fractions B, D, and F must not be zero ( 0 ), since division by ( 0 ) is not defined.

2. If the numerators of the fractions, A , C , or E are equal to zero ( 0 ), then the fractions, ( A / B ) = ( 0 ), ( C / D ) = ( 0 ), ( E / F )= ( 0 )since zero divided by any number, except zero, is equal to zero. That is ....
   ( 0 / B ) = ( 0 ), ( 0 / D ) = ( 0 ), and ( 0 / F ) = ( 0 ).
   Also, if A is a Negative Integer and B is a Positive Integer or A is a Positive Integer and B is a Negative Integer, then the Fraction ( A/B )
   is a Negative Fraction. If both A and B are Negative Integers then the Fraction ( A/B ) is a Positive Fraction.

3. If the number one ( 1 ) is divided by any integer, where Z is that integer, and Z is not equal to ( 0 ), then the fraction ( 1 / Z ) is called the RECIPROCAL of Z. Also if ( 1 ) is divided by any fraction, ( A / B ), where ( A / B ) is not equal to ( 0 ),
   then the result ( B / A ) is called the RECIPROCAL of ( A / B ).

4. Any integer multiplied by its' RECIPROCAL, the product is equal to 1. That is.... ( Z ) x ( 1 / Z ) = 1. Similarly,
   Any fraction multiplied by its' RECIPROCAL, the product is equal to 1.
   That is.... ( A / B ) x ( B / A) = 1.

5. 

   To MULTIPLY the fractions ( A / B ), ( C / D ), and ( E/ F )
   Multiply the numerators of the fractions, which is equal to ( A x C x E ).
   Then Multiply the denominators of the fractions, which is equal to
   ( B x D x F ). Write the product of ( A x C x E ) as the numerator of the answer and the product of ( B x D x F ) as the denominator of the answer. Please note if there are odd number of negative integers, then the answer to the product of this problem will be negative. If there are even number of negative integers, then the answer will be postitive. Please click on the Image for a better understanding.

6. In summary, to MULTIPLY the fractions ( A / B ) x ( C / D ) x ( E / F ), we do the following ... ( A / B ) x ( C / D ) x ( E / F ) = ( A x C x E ) / ( B x D x F ). If the fraction of the answer can be reduced to lowest terms, please do so and go to the Article written by this same author : How to REDUCE a fraction to lowest terms.

7. 

   To DIVIDE any three fractions ( A / B ), ( C / D ), and ( E / F ) it is very IMPORTANT to remember the order of operations. That is in the problem ( A / B ) ÷ ( C / D ) ÷ ( E / F ), first, the fraction ( A / B ) is Divided by the fraction ( C / D ) , then the result is divided by the fraction ( E / F ). To divide ( A / B ) ÷ ( C / D ), it is IMPORTANT to know which fraction is the Dividing Fraction, ( also called the DIVISOR ). So in order to DIVIDE the two fractions, we take the RECIPROCAL of the DIVISOR and MULTIPLY the First fraction. So the result is ( A x D ) / ( B x C ). That fraction is then divided by the fraction ( E / F ). The final answer will be ( A x D x F ) / ( B x C x E ). Please click on image for a better understanding.

8. In summary, DIVISION is MULTIPLYING by the RECIPROCAL, so to DIVIDE the fractions ( A / B ) ÷ ( C / D ) ÷ ( E / F ),we do the following ... ( A / B ) ÷ ( C / D ) ÷ ( E / F ) = ( A / B ) x ( D / C ) x ( F / E ) = ( A x D x F )/( B x C x E ). If the fraction of the answer can be reduced to lowest terms, then please go to the Article by the same author: How to REDUCE a fraction to lowest terms.

9. For example... Given the problem, MULTIPLY ( 15 / 16 ) x ( 8 / 9 ) x ( 4 / 5 ). The result is... ( 15 / 16 ) x ( 8 / 9 ) x ( 4 / 5 ) = ( 15 x 8 x 4 ) / ( 16 x 9 x 5 ) = ( 480 / 720 ). But ( 480 / 720 ) can be reduced to its lowest terms. So we reduce the fraction by expressing the numerator and the denominator in terms of Prime factors, that is... ( 480 / 720 ) = ( 2x2x2x2x2x3x5 )/( 2x2x2x2x3x3x5 ) = ( 2 / 3 ), since ( 2x2x2x2x3x5 )/( 2x2x2x2x3x5 ) = 1

10. For example...Given the problem, DIVIDE ( 15 / 16 ) ÷ ( 8 / 9 ) ÷ ( 4 / 5 ). The result is... ( 15 / 16 ) ÷ ( 8 / 9 ) ÷ ( 4 / 5 ) = ( 15 / 16 ) x ( 9 / 8 ) x ( 5 / 4 ) = ( 15 x 9 x 5 ) / ( 16 x 8 x 4 ) = ( 675 / 512 ). The fraction ( 675 / 512 ) is in lowest terms, since the numerator and the denominator when expressed in terms of prime factors have no common factors. That is ( 675 / 512 ) = ( 3x3x3x5x5 )/( 2x2x2x2x2x2x2x2x2 ).

Tips & Warnings

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