Addition and Subtraction of Rational Numbers

Lowest Common Denominator

Adding and subtracting fractions or rational numbers with identical denominators is simple. We add or subtract the numerators, then write the result over the denominator and, if possible, we factor to reduce the fraction to lowest terms as shown here:

note: in the 2nd example, we must put 1 in the numerator because we divided top and bottom of the fraction by (c – d). A common error is to cross out the c – d and assume that the numerator is now zero. Wrong!! It's 1!

In order to add or subtract rational numbers with different denominators, we must find their lowest common denominator so that we have pieces of the same size. What does this mean?

In a case where one demominator is a factor of the other, it's easy to get pieces of the same size.

Say you have half a dollar and a quarter of a dollar.

It's pretty obvious that a half = 2 quarters. So you have a total of 3 quarters of a dollar.
If you look in your pocket -- isn't that exactly what you have -- 3 quarters ???

Say we have two pizzas of the same size. The blue pizza was cut into 6 equal slices as usual, but the red pizza was cut into only 3 equal slices. We eat 5 slices of the blue pizza and 1 slice of the red one. So, there's 1 blue slice and 2 red slices remaining. Now we want to know how much of a whole pizza we have left.Which means we want to add (blue) and (red)

First, we find out how many sixths there are in the 2 remaining slices of the red pizza. This means we want the denominator of the fraction to be 6. To make the fraction's denominator 6, we must multiply 3 (its denominator) by 2, so we must also multiply its numerator by 2, since multiplying both numerator and denominator by the same number is multiplying by 1, which doesn't change the value of the fraction. We would proceed like this:

It's easy to see again, like in the example with the 3/4 of a dollar, that when we split a slice (a third) of the red pizza in half -- yes, each piece is half as big as it was -- but now there's 2 pieces. So we doubled the number of pieces at the same time as we cut the original piece in half. In the math and the reality, we multiplied and divided by 2.

note: If all denominators are primes, the lcd is the product of the primes.

for example: .

Now let's do some with variable denominators.

To find the lowest common denominator (lcd) for a number of fractions, factor the denominators of all the fractions in question, then choose the lowest common multiple of these factors.

For example,

First, we factor the two denominators and then find the lcd. The next step will change the two fractions into equivalent fractions with the same common denominator. Then we will subtract the numerators and write their difference over the common denominator. The process will go like this:

Note: The common denominator in this question is xy (x+y), so the first fraction has to be multiplied by and the second fraction has to be multiplied by .

Remember that we are looking for the lowest common denominator. A common error is to just use the product of all the denominators in the question instead of the lowest denominator possible. This not good since our objective is to do the question as simply and efficiently as possible.

For instance, say we wish to add the three fractions:

The lowest common denominator is x ⁴, not x ⁹, the product of the three denominators. Why x ⁴ ? Because each of the denominators in the question is a factor of x ⁴ . Here, the 1st fraction should be multiplied by

There are cases where we can reduce some of the fractions in the question to lowest terms before we find the lcd. This is why we should factor wherever possible before choosing the lcd.

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examples1

a)
	Factor the denominators. The lcd must have all factors of all the denominators.
	Multiply the 1st fraction by ,
	multiply the 2nd fraction by .
	Multiply, collect terms in the numerator.
b)
	Factor 1st denominator (difference of squares). reduce (x - 2) in top and bottom.
	The lcd is (x + 2).
	Subtract the numerators and put the difference over the lcd.
c)
	Factor and reduce within the fractions.
	The lcd is (x - 4)(x + 4).
	Multiply the numerators by the correct terms to create the lcd,
	remove brackets, collect like terms.

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.complex or split-level fractions

A complex (or split-level) fraction is one which has one or more fractions in either its numerator or denominator or both.

Here are some examples of complex fractions (or rational numbers):

The way to work with complex fractions is to realize that a fraction is just another way of indicating division, with numerator divided by denominator, so the three fractions seen above can be rewritten as division problems like this:

Now, all we have to do is to perform the operations in the brackets and then divide.

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examples-2:

1)	Rewrite the question as division. Find the lcd for each bracket. Invert the 2nd fraction. reduce the a's. Multiply.

2)	2) Rewrite the question as division. Find the lcd for each bracket. Invert the 2nd fraction. reduce the y's. Multiply.
3)	3) Rewrite the question as division. Find the lcd for each bracket. Invert the 2nd fraction and factor. reduce the like terms in the numerators and denominators and multiply.

Fraction Facts for Addition and Subtraction

1. Factor, reduce within the fractions wherever possible. Remember that reducing is done over multiplication and not over addition or subtraction.
2. Find the lowest common denominator for all fractions in the question. Remember to multiply both numerator and denominator by the same term so that you are multiplying by a fraction such as which has a value of 1.
3. Add or subtract the numerators, put the results over the lowest common denominator.
4. Factor, reduce the result to lowest terms.
5. Rewrite complex (split-level) fractions as division.

   efficiency hint: while writing your fractions, keep an eye out for ways to avoid redundancy. Don't rewrite . You'll just have to do it later so why not now?

   adding & subtracting	examples-1	complex fractions
   examples-2	practice	solutions

   practice

   Perform the indicated operations. Reduce your answers to lowest terms.

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   adding & subtracting	examples-1	complex fractions
   examples-2	fraction facts	solutions

   solutions

   Perform the indicated operations. Reduce answers to lowest terms.

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   adding & subtracting	examples-1	complex fractions
   fraction facts	practice	examples-2

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