Algebra I/Content/Working with Numbers/Adding Rational Numbers
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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:
- 3/10 + 2/10 = 5/10 = 1/2
Notice the simplification: five parts out of ten is the half of the parts. But it is not always that simple and, when we encounter another situation, we try to get back to this simple situation. You just need to know that multiplying the numerator and the denominator at the same time by the same number will not change the value of the fraction:
- 1/1 = 2/2 = 3/3 = "the whole cake"
- 1/2 = 2/4 = 3/6 = "half of the cake"
Knowing this, we can change the denominators of the fractions so that both denominators are the same. For example:
- 1/3 + 1/2 = (1x2)/(3x2) + (1x3)/(2x3) = 2/6 + 3/6 = 5/6
In this case we changed the fraction so all fractions have a denominator of 6.
Practice with simple fractions
Calculate the following additions:
- 1/2 + 1/4 = ...
- 1/3 + 1/6 = ...
- 1/4 + 1/6 = ...
(results: 3/4, 3/6=1/2, 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 !
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