How To Multiply Fractions With Whole Numbers

How To Multiply Fractions With Whole Numbers :

      Multiplication of two fractional numbers or rational numbers is  solved only in three steps i.e.

      Before we discuss the method of how to multiply fractions we have to understand that what is a fractional number or a rational number?

      So let us go through definition,

Definition Of Fraction or Rational Numbers :

              A fraction or rational number is of the form p/q, where p and q are integers and q≠0. 

         Then here is the question arises that what is an integer? 

       To answer this question let us take examples,

     The set of numbers like .............-3,-2,-1,0,1,2,3..........

     are called as the set of integers. 

      Now come to our initial question of how to multiply fraction with mixed numbers?

    Let us discuss the methods

(1) First of all multiply the numerators of two fractional numbers i.e.

for example     3 / 5   ×   2 / 5 

    Here first we have to multiply 3  with 2.

        Let us proceed another step 

(2) Then multiply the same or different  denominators of the two fractional numbers 

 for example    3 / 5  × 2  / 5 

      Here to multiply  5 of the fraction 3/5 with    5  of the fraction 2/5. 

 (3) Here is the next step to simplify if needed ,

     for example   3 × 2 / 5 × 5 = 6 / 25 

       

           Here we know that there is nothing to simplify. 

      But there are some problems that needed simplification after multiplication for an accurate result. 

        For example 2 / 4 × 6 / 8 = 2 × 6 / 4 × 8 

                                            =  12 / 32 

       Here we need simplification, we have to simplify the fractional number till numerator or denominator is converted into prime numbers, i.e.

         Let us take the above example that 

       12 / 32, where the numerator is 12 . Let us divide it with such number which results in a prime number like 2,3,5,7,11...etc. 

          Let us divide it by 4, then the result is 12/ 4 = 3 which is a prime number.

       There is a rule of the fraction that if we divide or multiply some number with numerator we have to divide or multiply the same whole number or any integer with denominator.

     So let us divide the denominator by 4 we get  32/4 = 8 

    Hence the simplification result is 3/8. 

      We reached a conclusion that 

           A rational number is of the form p/q, where p and q are integers and q ≠ 0  and p and q both are relatively prime. 

        One thing we have to consider that in case of complicated multiplication or multiply two large numbers is not so easy.

       So, in that case, we have to simplify the fractions before multiply with each other, 

      For example :

                  374 / 462  ×  564 / 126

    Ans:  In this case, we have to first simplify   374 / 462 to relatively prime numbers.

         In case we can't find the exact number which if we divide by 374 it is converted into a prime number, then divide this number with a least positive divisor by which it is divisible. 

      So let us divide both 374 and 462 by 2 we get 187 / 231, so here we get the exact relatively prime numbers. 

      Let us do the same with a second fraction that 564 / 126. 

      Let us divide both 564 and 126 by 2 we get  282/63. Again let us divide the result by 3 we get 94/21. Here we get the relatively prime numbers. 

    So let us multiply 

             187/231 ×  94/21 

        =   187 × 94 / 231 × 21 

        =   17578 / 4851 

      which is the required solution.
       This is the solution to the problem How to multiply an improper fraction with a proper fraction.

             Let us discuss the improper and proper fraction .

Improper Fraction:

            The fraction of the form p/q is said to be an improper fraction if p> q . 

for example :

                   3/2 , 4/3 , 5/4 , 7/6........etc.

Proper Fraction :

           The fractional number of the form p/q is said to be a proper fraction if p<q . 

     for example :   3/4 , 11/12, 16/17, 49/50 , 60/61 ............etc .

       

Prime Number :

         The number which is only divisible by 1 or itself is called a prime number.

  Ex :  2,3,5,7,11,13,17,19,.......

Relatively Prime :

       Two numbers are called relatively prime to each other if their greatest common divisor or g.c.d is 1.

      This is all about How To Multiply Fraction.

References:-

(I).  Multiplication And Fractions Math  Games tough topics.

(II). Multiplication, division, the addition of fractions by OUP Oxford.

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