Basic Algebra/Working with Numbers/Adding Rational Numbers

Vocabulary

Numerator
Denominator
Irreducible

Lesson

It is easy to add fractions when the denominators are equal. For example, adding ${\frac {3}{10}}$ and ${\frac {2}{10}}$ is very simple, just add the numerators and you have the numerator of the resulting fraction:

${\frac {3}{10}}+{\frac {2}{10}}={\frac {5}{10}}={\frac {1}{2}}$

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:

${\frac {2}{2}}={\frac {3}{3}}={\frac {107}{107}}=1$

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

${\frac {1}{3}}+{\frac {1}{2}}={\frac {1\times 2}{3\times 2}}+{\frac {1\times 3}{2\times 3}}={\frac {2}{6}}+{\frac {3}{6}}={\frac {5}{6}}$

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

More complicated fractions

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

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 :

${\frac {123}{456}}+{\frac {234}{120}}={\frac {123\mathbf {\times 120} }{456\mathbf {\times 120} }}+{\frac {234\mathbf {\times 456} }{120\mathbf {\times 456} }}={\frac {14760}{54720}}+{\frac {106704}{54720}}={\frac {14760+106704}{54720}}={\frac {121464}{54720}}$

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:

${\frac {1}{6}}+{\frac {1}{4}}={\frac {1\times 2}{6\times 2}}+{\frac {1\times 3}{4\times 3}}={\frac {2}{12}}+{\frac {3}{12}}={\frac {5}{12}}$

We only multiplied by 2 in the first fraction and by 3 in the second fraction. The resulting fraction, ${\frac {5}{12}}$ is optimal, which we call irreducible.

Note that 2 is the half of $4=2\times 2$ and 3 the half of $6=3\times 2$ . 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=\mathbf {3} \times 41

and

456=2\times 228=2\times 2\times 114=2\times 2\times 2\times 57=2\times 2\times 2\times \mathbf {3} \times 19

234=2\times 117=2\times 3\times 39=\mathbf {2\times 3} \times 3\times 13

and

120=2\times 2\times 3\times 10=2\times \mathbf {2\times 3} \times 2\times 5

We can see that we can simplify ${\frac {123}{456}}$ by 3 which gives ${\frac {41}{2\times 2\times 2\times 19}}$ and simplify ${\frac {234}{120}}$ by 2 × 3 which gives ${\frac {39}{2\times 2\times 5}}$ . 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 $2\times 2\times 2\times 19$ and the factors $2\times 2\times 5$ . It is the number that has just all these factors in correct number: $2\times 2\times 2\times 5\times 19=760$ .

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

${\frac {123}{456}}+{\frac {234}{120}}={\frac {41}{2\times 2\times 2\times 19}}+{\frac {39}{2\times 2\times 5}}={\frac {41\mathbf {\times 5} }{760}}+{\frac {39\mathbf {\times 2\times 19} }{760}}={\frac {205}{760}}+{\frac {1482}{760}}={\frac {1687}{760}}$