Common Denominator: Pieces of the Same Size

We know that the sum of ½ and ¼ is ¾ because we know that ½ = 2 × ¼.
So ½ + ¼ is exactly the same as (2 × ¼) + (1 × ¼) which is 3 × ¼ or ¾.
The common denominator for ½ and ¼ is 4 since both denominators are factors of 4 and we know that ½ is exactly the same as 2 quarters. In other words, 2-quarters and ½ are equivalent fractions.

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 ???

they must have the same denominator.

Adding Fractions with a Common Denominator

It is easy to add or subtract fractions (or rational numbers) that have the same number in their denominators, because they already have a common denominator. We just add or subtract their numerators, then write the result over the denominator and, if possible, we reduce the fraction to lowest terms -- like this:

note: in the 1st example, we reduced 10/5 to 2 by division.

In order to add or subtract fractions with different denominators, we determine their lowest common denominator, then create equivalent fractions with this 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 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.

 The lowest common denominator (lcd) of fractions is the smallest number divisible by all the denominators of the fractions.

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

for example:

.

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. The next step changes all the fractions into equivalent fractions with the same common denominator. For example,

.

Here, the lowest common denominator is 3 × 2 × 5 = 30

The first fraction gets multiplied by 5/5, the second by 2/2.

We get .

Remember, we're 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, in the question above, had we not factored the denominators to see the common factor of 3, we would have used 90 as the lcd instead of 30.

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.

For example: .

Notice how we changed 3/27 into 1/9.

Whole Numbers (Integers) as Fractions

Every whole number or integer is a fraction too, but because the denominator is 1 (one), we don't write it. However, when we want to add or subtract whole numbers and fractions, we have to change them into fractions with a common denominator. Since we can only multiply by something equal to 1 so that we don't change any values, we choose the correct expression of a/a to multiply by.

For example:

.

and

.

Note how we use the proper expression for 1 in each case.
Since we needed thirds in the first example, we multiplied the whole number (4) by 3/3.
In the second case, we multiplied by 5/5 since we needed fifths.

Working with Mixed Numbers

A Mixed Number has 2 parts -- a whole number part and a fraction. To change a mixed number into an improper fraction, we change the whole number part to an equivalent fraction with the correct denominator, then we add the 2 fractions together like this:

So, when we need to add or subtract mixed numbers, this is the approach we use. We turn them into improper fractions with a common denominator (lowest one!) -- then we add or subtract them as instructed by the question.

Since the common denominator of 8 and 4 is 8, we multiply the second fraction by 2/2 in order to make the denominator 8 instead of 4. Then we add the numerators to get:

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice

1) Rewrite these whole numbers as fractions with the denominator shown:

 a) b) c) d) e)

(solutions).

Change answers that are improper fractions into mixed numbers.

 a) b) c) d) e)

(solutions).

Change improper fractions in the answers into mixed numbers.

 a) b) c) d) e) f) g) h)

(solutions).

Solutions

1) Rewrite these whole numbers (integers) as fractions with the denominator shown:

 a) b) c) d) e)