How to Multiply and Divide Fractions

UNH Mathematics Center

1  Multiplication and Division Rules

We'll begin with multiplication. The product of two fractions is a new fraction: we get its numerator by multiplying the numerators of the fractions we have, and we get its denominator by multiplying their denominators.

A B · C D = AC BD

Of course a fraction can also be multiplied by a number that isn't a fraction. Or maybe we should say, that doesn't look like a fraction? Because:

• A fraction might be multiplied by a number presented to us as a decimal. For instance: because 2.5 and 5/2 are two names for the very same number, we can write
  

  3 5 · 2.5 = 3 5 · 5 2 = 3·5 5·2 = 15 10

• Any number might be thought of as a fraction, with 1 as an ''invisible denominator.'' To multiply a fraction by a number that doesn't look like a fraction, we multiply the fraction's numerator by that number. For example,

  
  

  5· 3 4 = 5 1 · 3 4 = 15 4 .

Division is another form of multiplication. Just as dividing by 3 is the same as multiplying by 1/3; and, dividing by 1/3 is the same as multiplying by 3 ... to divide by a fraction is the same as multiplying by its reciprocal.

If the fractions in question are A/B and C/D the division will produce a new fraction:

A B C D = A B · D C = AD BC

For instance:

3 5 - 1 2 = 3 5 -1 2 = 3 5 · 2 -1 = 3·2 (5)(-1) = 6 -5 = - 6 5

1.1  Some Parentheses are Invisible

Sometimes students recognize the distributive property only as a rule to be remembered in a formula ... let's say, A(B+C) = AB+AC. It can happen that the student, mindful only of the formula, does not even notice that multiplication and division are interacting, unless the parentheses are obvious on the page.

But, when a fraction is to be multiplied by another number as in the example just above, the very appearance of a sum in the fraction's numerator is enough to tell us that the whole numerator is to be multiplied. We don't need parentheses to show us what is being multiplied. The entire numerator is being multiplied, and parentheses are not needed to make the point clear. In our example,

2x· 5-x+y2 x+35 = 2x 1 · 5-x+y2 x+35 = (2x)(5-x+y2) (1)(x+35) = 10x - 2x2+2xy2 x+35

Be particularly sure you notice uses of the distributive rule when minus signs are involved. This is a form of the same mistake that is easy to make over and over: it can get dismissed as ''a little minus sign mistake'' every single time, so that the student never realizes that the mistake is actually a failure to distribute multiplication over addition.

Here's an example:

(-x) y-x y-3 = -x 1 · y-x y-3 · (-x)(y-x) (1)(y-3) = -xy+x2 y-3

Look this over carefully. We've put in all the steps, just to make the point! Make sure you follow the progress of the minus sign in -x all the way through the multiplication. (The typical mistake is not to see the effect of this minus sign on the numerator's second term.)

2  Undoing Multiplication: Practice Safe Cancellation!

Canceling - used legitimately in fractions - is a way of using the multiplication rule ''backwards'' so as to simplify the fraction's appearance by rewriting it as a product of several factors, some of which have the value 1.

We can do this when a fraction's numerator and denominator have factors in common. The multiplication rule lets us rewrite our fraction as a product of several fractions. Those factors in the product that have value 1 can be ''eliminated'' in the sense that multiplying any number by 1 leaves it unchanged.

For example, the first fraction we looked at in this section had value 15/10. We are able to rewrite 15/10 as 3/2, because we can write it as a product in which one factor has value 1:

15 10 = 5·3 5·2 = 5 5 value 1  · 3 2 = 1· 3 2 = 3 2

We would do the same thing with an algebraic fraction. For instance, the expression

(x+2)2(y+1) (x+2)(y+1)2

can be legitimately rewritten by cancellation as

x+2 y+1 .

This is because we can rewrite it so that it displays factors with the value 1:

(x+2)2(y+1) (x+2)(y+1)2 = x+2 x+2 this is 1  · x+2 1 · y+1 y+1 this is 1  · 1 y+1 .

The way some students get into trouble with cancellation is by concentrating on simplifying the appearance of the algebraic expression on the page, instead of thinking of the procedure as an occasion for ''multiplying-by-one.'' The error comes in when you think of  ''canceling'' as eliminating duplicate instances of the same symbols wherever they appear in an expression

Legitimate cancellation, when done with algebraic fractions, always amounts to rewriting the fraction so that it displays factors that are equal to 1. This is because 1 is multiplication's ''neutral'' number: numbers are unchanged when multiplied by 1. If you cannot rewrite a fraction in this way, the ''canceling'' you are doing is probably wrong!

2.1  In case of error: check here first

tidying up

This happens to the person who tries to tidy up expressions by looking for identical items within them. Whenever identical items are spotted within the expression, the person simply zaps them away with diagonal lines.

playing fair

This happens to the person who adopts a fair-play attitude with respect to a fraction's numerator and denominator, doing the same thing to each.

2.1.1  The tidying-up error

2.1.2  The fair-play error

Surely it's inherited from the plan of doing the same thing with both sides of an equation, which is the right thing to do. But the reason we do this with equations is that the expressions on the left and right sides of the equals sign represent the same number, and we mean to keep it that way. So it makes sense that whatever we do to the expression on the left side of the equals sign, we'll do that same thing to the expression on the right side.

Fractions are a different matter. There was never the least idea that a fraction's numerator, and its denominator, necessarily represent the same number! The important thing about a fraction is that the ratio of its numerator and denominator remain unchanged. So, the only legitimate same-thing we can do to a fraction's numerator and denominator is to multiply (or divide) each of them by the same nonzero number. Other do-the-same-thing plans will almost certainly get you into the soup.

For instance: if we subtract 3 from both numerator and denominator of

x+3 2x+3 ,

we will get a fraction whose value is 1/2 (for any number x except zero). You can see, that makes no sense at all (try it for x = -3 or for x = 0, for instance).

3  Two After-Words

3.1  About Zero

The other ''neutral element'' in the family of numbers is 0. Adding and subtracting 0 will not change a number's value. But 0 is a number with a sting: if you multiply anything by 0 the product is 0. Not to put too blunt an edge on it; but 0 annihilates numbers.

Here's what this means for fractions:

• Fractions whose numerators are zero, have the value zero.
• ... And it works the other way too. The only way a fraction can have the value zero, is for its numerator to be zero!
• ''Fractions whose denominators are zero'' are not really numbers at all. A thing that looks like this isn't a number, and doesn't have a value.
• We don't even want to imagine a thing that looks like a fraction, but has both numerator and denominator zero. Such a thing is to be avoided at all costs. It surely bites, and is probably rabid.

3.2  Simple means Useful

We think a fractional expression is most useful to the reader if its denominator is left in factored form. The expression is more easily interpreted that way (we might want to know which values of a variable make the denominator zero, for instance), so multiplying out the denominator not only is extra work but usually makes the answer less useful to you.

4  Problems for You to Work

1. Here are some fractions for you to multiply or divide. In each case write the result in an appropriately simple form. Display explicitly any ONE-valued fractions if you cancel.
   1. 
      

      x y · -3y x2 · y+4 x-2

   2. 
      

      (-6)·(x-2)· 3-x x+2 ·4· 1 (2-x)2

   3. 
      

      x-3 y -2x+6 y3(3-x)

2. Reduce the algebraic fraction below to its simplest possible expression. You will want to factor the numerator and denominator first. Can you tell, without doing any explicit numerical evaluation, whether it is positive or negative if
   -2 < x < -1?
   

   x5-x3 x2+3x+2

3. 1. Factor the numerator and denominator of this fraction (you can factor the denominator by grouping), so that you can simplify it as much as possible.

      
      

      -5x2-5x+10 (x4+x)-(x3+1)

   2. Here's one that might be just a bit more challenging. It will probably help you to know that x³+1 can be factored. One of its factors is x+1; if you don't remember the other one, you can always find it by using long division. Factor and simplify the expression
      

      x2-1 (x4+x)-(x3+1)