Expert: Josh - 2/21/2006

Question
3k / k-2 + 6 / 2-k?

I don't even know where to start. Could you please help. Thanks>

Answer
Fatima,

It is really important that you put brackets around expressions when you are writing fractions in a single line; otherwise, they can be easily misinterpreted.

I presume your question reads
3k/(k-2) + 6/(2-k)
we are asked to simplify this expression and work out the LCD (least common denominator) on the way.

What does the Least Common Denominator mean?
MOTIVATION:
Consider the two fractions in (1/3) + (1/4). We pay attention to the denominator (bottom part under the fraction bar). We have 3 and 4 respectively. Clearly, we cannot add (1/3) and (1/4) directly, because one part in three is NOT the same as one part in four. The "one" differ in meaning. In general, to simplify expressions involving fractions, we need to make sure that the numerator parts are expressed relative to a common denominator. In this example, 1/3 is equivalent to (1/3)*(4/4)=4/12. Similarly, 1/4 is equivalent to (1/4)*(3/3)=3/12. Observe that we haven't changed the proportions at all, but both fractions are expressed relatively to "parts in twelve". That is, they have a common denominator, this being 12. Now, it makes sense to add (4/12) with (3/12). This is the same as adding (1/3) with (1/4), and the answer is (4+3)/12 = 7/12.

We could have picked (1/3)=8/24 and (1/4)=6/24 instead, and add these together, to get 14/24. But 24 is NOT the smallest common denominator.

In your question, you've got an algebraic expression containing an unknown quantity "k". Don't worry about it, you manipulate the fractions in exactly the same way.

Note: There is a procedure for finding the LCD, but until you are more comfortable with this, I'd rather that you find this by trial and error, using intuition for simple exercises.

eg., LCD of 1/4 and 1/11 is 4*11 = 44
    LCD of 1/2 and 1/5 is 2*5 = 10
    LCD of 1/4 and 1/6 is 4*6/2 = 12 (N.B. not 24)

ANSWER TO YOUR QUESTION:
In 3k/(k-2) + 6/(2-k), observe that (k-2)=-(2-k),
so, 3k/(k-2) + 6/(2-k) = 3k/(k-2) + 6/-(k-2)
=3k/(k-2) - 6/(k-2) ...now we have the least common denominator (k-2), equally 2-k would be okay
=(3k-6)/(k-2)
=3*(k-2)/(k-2) ...cancel common factors
=3