# Basic Algebra/Working with Numbers/Adding Rational Numbers

## Contents

## Vocabulary[edit]

- numerator
- denominator
- irreducible

## Lesson[edit]

It is easy to add fractions when the *denominators* are equal. For example. adding 3/10 and 2/10 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.

## Practice with simple fractions[edit]

Calculate the following additions:

(results: )

## More complicated fractions[edit]

In these cases, we can guess which multiplication to do, but sometimes, it is not that easy. For example, adding 123/456 and 234/120.

- 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 :

- 123/456 + 234/120 = (123
**x120**)/(456**x120**) + (234**x456**)/(120**x456**) - = 14760/54720 + 106704/54720 = (14760 + 106704)/54720 = 121464/54720

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:

- 1/6 + 1/4 = (1 x2)/(6 x2) + (1 x3)/(4 x3) = 2/12 + 3/12 = 5/12

We only multiplied by 2 in the first fraction and by 3 in the second fraction. The resulting fraction, 5/12 is optimal, which we call **irreducible**.

Note that 2 is the half of 4=2x2 and 3 the half of 6=3x2. 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...

- 123 =
**3**x41 and 456 = 2x228 = 2x2x114 = 2x2x2x57 = 2x2x2x**3**x19 - 234 = 2x3x39 =
**2x3**x3x13 and 120 = 2x2x3x10 = 2x**2x3**x2x5

We can see that we can simplify 123/456 by **3** which gives 41/(2x2x2x19) and simplify 234/120 by **2x3** which gives 39/(2x2x5). 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 2x2x2x19 and the factors 2x2x5. It is the number that has just all these factors in correct number : 2x2x2x5x19 = 760.

To attain this number, we must multiply in the first fraction by **5** and in the second by **2x19**. So, finally we have:

- 123/456 + 234/120 = 41/(2x2x2x19) + 39/(2x2x5) = (41
**x5**)/760 + (39**x2x19**)/760 - = 205/760 + 1482/760 = 1687/760

This fraction is simpler as the first obtained 121464/54720.

Both fractions are equal : 1687/760 = 121464/54720

But the factor between the two fractions is 72 !

## Practice Games[edit]

put links here to games that reinforce these skills

## Practice Problems[edit]

(Note: put answer in parentheses after each problem you write)