is cancelling a shortcut for adding?

Question

Is cancelling a shortcut for adding fractions?

Answer 1

You can if you are referring to finding a number that "goes into" the numerator (n) & the denominator (d) in one fraction evenly; this is commonly called "reducing". This is beneficial since you have to have the same (d) in all fractions to add them. You can't "cancel" one number from the 1st fraction, and a number from the 2nd fraction. I think the "cancelling" you're referring to is what's done when you multiply fractions.

Answer 2

Cancelling fractions is generally the term used when reducing fractions that are being multiplied, not added.

To add fractions we have to get a common denominator... same number on the bottom of each fraction.  This

is done by finding the least common multiple of the denominators then adding the fractions.

 

The least common multiple (LCM) is the smallest number that both will divide into.  For example, the LCM for 2 and 3 is 6.  6 is the smallest number that both 2 and 3 will divide into.  Thus to add 1/3 + 1/2 we need to convert both to sixth's.

 

1/3 + 1/2 =

 

To convert 1/3 to x/6 multiply by 2/2:   (1/3)(2/2) = 2/6

To convert 1/2 to x/6 multiply by 3/3:   (1/2)(3/3) = 3/6

 

1/3 + 1/2 = 2/6 + 3/6 = (2+3)/6 = 5/6

 

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For cancelling we are discussing multiplying fractions;

If we have common factors in the numerator and denominators

of fractions that are being multiplied we can use cancellation

to reduce the fractions prior to multiplying them:

 

(4/9)(3/2)

The 2 in the denominator of the 2nd fraction can be cancelled into

the 4 in the numerator of the 1st fraction giving:

 

(4/9)(3/2) = (2/9)(3/1)

 

Now we can cancel the 3 in the numerator of fraction 2 into the 9

in the denominator of fraction 1 giving:

 

(2/9)(3/1) = (2/3)(1/1) = 2/3

 

So:  (4/9)(3/2) = 2/3

 

Check:  (4/9)(3/2) = (4)(3)/(9)(2) = 12/18 = (6)(2)/(6)(3) = 2/3