How to MULTIPLY or DIVIDE any two Fractions QUICKLY

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 two fractions QUICKLY.

Things You'll Need

• Pencil and
• Paper

Instructions

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

2. If the numerators of the fractions, A or C are equal to zero ( 0 ), then the fractions, ( A / B ) = ( 0 ), or ( C / D ) = ( 0 ), since zero divided by any number, except zero, is equal to zero. That is ....
   ( 0 / B ) = ( 0 ), and ( 0 / D ) = ( 0 ).

3. If the number one ( 1 ) is divided by any integer, say Z is that integer, and Z must not be equal to ( 0 ), then the fraction ( 1 / Z ) is called the RECIPROCAL of Z. Also if ( 1 ) is divided by any fraction,
   say ( A / B ) is that fraction, and ( A / B ) must not equal to ( 0 ),
   then the fraction ( 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 ) and ( C / D )
   Multiply the numerators of the fractions, which is equal to ( A x C ).
   Then Multiply the denominators of the fractions, which is equal to
   ( B x D ). Write the product ( A x C ) as the numerator of the answer and the product ( B x D ) as the denominator of the answer.

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

7. To DIVIDE any two fractions ( A / B ) and ( C / D ), it is very IMPORTANT to know which fraction is the Dividing fraction, that is in the problem (( A / B )) / (( C / D )), the fraction ( C / D ), 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.

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

9. For example... MULTIPLY ( 15 / 16 ) x ( 8 / 9 ).
   ( 15 / 16 ) x ( 8 / 9 ) = ( 15 x 8 ) / ( 16 x 9 ) = ( 120 / 144 )
   but ( 120 / 144 ) can be reduced to lowest terms. So we reduce the fraction by expressing the numerator and the denominator in terms of Prime factors, that is...
   ( 120 / 144 ) = ( 2x2x2x3x5 )/( 2x2x2x2x3x3 ) = ( 5 / 6 ), since
   ( 2x2x2x3 )/( 2x2x2x3 ) = 1

10. For example... DIVIDE (( 15 / 16 )) / (( 8 / 9 )).
    (( 15 / 16 )) / (( 8 / 9 )) = ( 15 / 16 ) x ( 9 / 8 ).
    ( 15 / 16 ) x ( 9 / 8 ) = ( 15 x 9 ) / ( 16 x 8 ) = ( 135 / 128 ).
    The fraction ( 135 / 128 ) is in lowest terms, since the numerator and the denominator when expressed in terms of prime factors have no common factors. That is ( 135 / 128 ) = ( 3x3x3x5 )/( 2x2x2x2x2x2x2 ).

11. Another method is...to first find the common prime factors in the numerators and the denominators of the fractions and reduce to lowest terms, then do the MULTIPLYING of... ( 15 / 16 ) x ( 8 / 9 ), that is
    ( 15 / 16 ) x ( 8 / 9 ) = ( 15 / 9) x ( 8 / 16 ) = ( 5 / 3 ) x ( 1 / 2 )
    ( 5 / 3 ) x ( 1 / 2 ) = ( 5 x 1 ) / ( 3 x 2 ) = ( 5 / 6 ).

Tips & Warnings

• To MULTIPLY or DIVIDE any two fractions in less time than the method given,

• Look carefully at the numerators and the denominators of both fractions

• See if any of the numerators have common prime factors with any of the denominators and reduce to lowest terms.

• The Answer will be the ( product of the remaining factors of the numerator) divided by the ( product of the remaining factors of the denominator).

• Please NOTE, that in the operation of the DIVISION of two fractions, the fraction that follows the division sign, the RECIPROCAL of that fraction must be taken, and then multiply the fraction that preceeds the division sign by it.