Basic Algebra/Working with Numbers/Adding Rational Numbers

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Vocabulary

numerator
denominator
irreducible

Lesson

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:

{\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.

Practice with simple fractions

Calculate the following additions:

${\frac {1}{2}}+{\frac {1}{4}}=...$
${\frac {1}{3}}+{\frac {1}{6}}=...$
${\frac {1}{4}}+{\frac {1}{6}}=...$

(results: ${\frac {3}{4}},{\frac {3}{6}}={\frac {1}{2}},{\frac {5}{12}}$ )

More complicated fractions

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 = 3x41 and 456 = 2x228 = 2x2x114 = 2x2x2x57 = 2x2x2x3x19

234 = 2x3x39 = 2x3x3x13 and 120 = 2x2x3x10 = 2x2x3x2x5

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) = (41x5)/760 + (39x2x19)/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

put links here to games that reinforce these skills

Practice Problems

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

${\frac {1}{8}}+{\frac {1}{4}}=({\frac {3}{8}})$
${\frac {1}{8}}+{\frac {1}{2}}=({\frac {5}{8}})$
${\frac {2}{3}}+{\frac {1}{9}}=({\frac {7}{9}})$
${\frac {2}{4}}+{\frac {1}{2}}=(1)$
${\frac {8}{8}}+{\frac {1}{2}}=(1{\frac {1}{2}})$

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