2.04   Multiplying and Dividing Fractions

Intermediate Algebra: One Step at a Time    Page 157-164:  #19, 24, 25, 26, 28, 30, 31, 32, 35, 39, 40

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

19.                     

This is a division problem, so you must invert the second fraction and multiply.  Several factors will divide out.  Notice that these are FACTORS, not TERMS.

First, 14 divides into 42 and it goes 3 times, and 9 divides into 27, and it goes 3 times.   

Next, the 3 factors divide out: 

Now, the  factor divides out with the  leaving an factor in the numerator, and the  factor divides out with the  leaving a factor in the numerator.  It looks like this:

The final answer is  or

24.                  

This is a multiplication problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.

Notice that the (x-1) in the first numerator and the (x-1) in the second denominator divide out.

Also the (x-4) in the first denominator and the second numerator divide out. 

And the (x+1) in the second numerator and denominator divide out.  

What is left is the x in the numerator and the (x+3) in the denominator.  

Final Answer:                

25.                  

This is a multiplication problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.

Notice that the (x+7y) in the first numerator and the (x+7y) in the first denominator divide out.

Also the (x+2y) in the second numerator and the second denominator divide out. 

And the (x-7y) in the first numerator and second denominator divide out.  

What is left is the (x-5y) in the numerator and the (x+5y) in the denominator.  

                    

26.                    

This is a multiplication problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.

Notice that one of the (x-4y) factors in the first numerator and the (x-4y) in the second denominator divide out.

Also the (x+2y) in the second numerator and the (x+2y) in first denominator divide out. 

The final answer is .

If you want to do so, you can multiply out the numerator and denominator, but this is not necessary.  The factored form is actually preferred!

28.                      

This is a division problem, so you must invert the second fraction and multiply.  Meanwhile, you should try to factor everything that you can, in order to set up the next step of dividing out factors.  NEVER DIVIDE OUT TERMS!!  In the first denominator, there are common factors of 6 and x to take out.  The second denominator is a difference of two squares which factors and becomes the second numerator.

Notice that the (y+2) in the first denominator and the (y+2) in the second numerator divide out.

Also the in the first numerator divides out with the  in the second denominator, leaving a   in the denominator. 

Divide out the x factor, leaving

The final answer is

30.              

This is a division problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.  In fact, the hardest part of this problem will be the factoring.  If you have trouble factoring this one, then check out these links: 

                    Advanced Factoring of Trinomials in Living C O L O R;

                    One Step Explanation, Examples, Exercises, and Answers.

Notice that the in the first numerator and denominator divide out, and the   factor in the first numerator and the in the second denominator divide out.

Also, notice that the  in the numerator divides out with the   of the denominator, leaving an in the denominator. 

This is the final answer: 

31.       

This is a division problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.  In fact, the hardest part of this problem will be the factoring.  If you have trouble factoring this one, then check out these links: 

                    Advanced Factoring of Trinomials in Living C O L O R;

                    One Step Explanation, Examples, Exercises, and Answers.

Notice that the in the first numerator and denominator divide out.  The   factor in the first denominator and the in the second numerator divide out.  Finally, the  in the second numerator and denominator divide out.

Final Answer:                 

32.        

This is a division problem, so you must factor, factor, factor, and factor!  Factor everything, and see if any factors divide out.  In fact, the hardest part of this problem will be the factoring.  If you have trouble factoring this one, then check out these links: 

                    Advanced Factoring of Trinomials in Living C O L O R;

                    One Step Explanation, Examples, Exercises, and Answers.

Notice that the in the first numerator and denominator divide out.  The   factor in the first numerator and the in the second denominator divide out.  Finally, the  in the second numerator and denominator divide out.

Final Answer:                 

35.                     

This is a division problem, so you must invert the second fraction and multiply.  Factor whenever possible in order to reduce the fractions in the next step.

Notice that the (5-x) in the first numerator is the negative of the (x-5) in the second denominator.  A number divided by its negative is -1, so these factors divide out, leaving a -1 in the first numerator.

Also, the 4 divides into the 12 leaving a factor of 3 in the numerator, and the x and the y divide out, leaving  in the denominator and  in the numerator

Final Answer:             

 39.                    

This is a multiplication problem, so of course you must factor whenever possible in order to reduce the fractions in the second step.  However, the factoring might be tricky here.  Notice that in the first fraction, the numerator is a difference of cubes, and the denominator requires factoring by grouping.  In the second fraction, the numerator is already given in a factored form, and the denominator does NOT factor at all.  In this problem, you must factor the first fraction, but leave the second fraction alone!

At this point, notice that the denominator is NOT factored yet, so continue by taking out the common factor of .

Now, notice that in the first denominator  is a difference of two squares, so this factors again.

Next, the factors indicated below in green divide out:

What is left when these factors are divided out is the final answer.

Final Answer:                                     

40.                                  

This is a division problem, so of course you must factor whenever possible in order to reduce the fractions, and invert the second fraction and multiply.  However, the factoring might be tricky here.  Notice that in the first fraction, the denominator is a difference of cubes.  In the second fraction, the numerator is a perfect square trinomial, and the denominator is already given in a factored form.

You might want to write out the second numerator as

 .

Next, the factors indicated below in green divide out:

What is left when these factors are divided out is the final answer.

Return to main page        Math in Living C O L O R !!

     Return to Basic Algebra page     Return to Intermediate Algebra page  

 Return to College Algebra page