# Basic Algebra/Working with Numbers/Adding Rational Numbers

*5 October 2017*. There are 4 pending changes awaiting review.

## Vocabulary[edit]

- Numerator
- Denominator
- Irreducible

## Lesson[edit]

It is easy to add fractions when the **denominators** are equal. For example, adding and is very simple, just add the **numerators** and you have the numerator of the resulting fraction:

Notice the simplification: five parts out of ten is the half of the parts.
Unfortunately, it is not always so simple. Sometimes we need to add fractions that have **different** denominators. Before we can add them, we must alter the fractions so that their denominators are the **same**. We can do this by multiplying each fraction by the number one which doesn't change the value of the fraction). However, the *form* of the number one will itself be represented as a fraction whose denominator and numerator are equal, and under our control. For example, all of these fractions are equal to one:

Knowing this, we can change the denominators of the fractions so that the denominators of both are the same. For example:

In this case we changed both fractions so that they each had a denominator of 6.

## More complicated fractions[edit]

In these cases, we can guess which multiplication to do, but sometimes, it is not that easy. For example, adding and .

The simplest general method is to multiply the numerator and denominator of the first fraction by the denominator of the second fraction and vice-versa. The resulting denominators will both be the product of the two original denominators.
In this case :

We obtain generally big numbers which is not optimal because the fraction can most of the time be written with smaller numbers.

The second is more subtle. Instead of multiplying by the actual denominators, we multiply by the **smallest possible number** for each side so that we obtain the same denominator. For example:

We only multiplied by 2 in the first fraction and by 3 in the second fraction. The resulting fraction, is optimal, which we call **irreducible**.

Note that 2 is the half of and 3 the half of . We did not multiply by the given denominators, we avoided to multiply by the factor 2. Let's take the previous example and find the factors composing the numbers...

- and
- and

We can see that we can simplify by 3 which gives and simplify by 2 × 3 which gives . Remember that multiplying by the same number the numerator and the denominator does not change the value. The same is true when dividing by the same number.

Now comes a question : which is the smallest integer that contains the factors and the factors . It is the number that has just all these factors in correct number: .

To attain this number, we must multiply in the first fraction by 5 and in the second by 2 × 19. So, finally we have:

This fraction is simpler as the first obtained .

Both fractions are equal:

But the factor between the two fractions is 72!

## Practice Games[edit]

put links here to games that reinforce these skills

## Practice Problems[edit]

Use `/`

as the fraction line and put spaces between the wholes and fractions!