Arithmetic with fractions
Equivalent fraction
Two or more fractions are equivalent when they represent the same quantity, and are written differently. For example, the fractions , , and are equivalent, since they represent the quantity "one half." Two fractions are equivalent if they can be obtained from each other by multiplying (or dividing) the numerator and the denominator by the same number, that is, by one.
Example:.
One way to know if two fractions and are equivalent is to check if they are equal: the fractions are equivalent if (equality is obtained by operating on ).[6].
In this way, equivalent fractions are reducible, since the numerator and denominator are not prime to each other and can be simplified into an irreducible fraction, in which the numerator and denominator are prime to each other. The set of all fractions equivalent to a given fraction is called a rational number, and is usually represented by the only irreducible equivalent fraction of the set. A specific case is when the numerator is a multiple of the denominator, then, by reducing it, any number belonging to the set of integers is obtained, which is why it is called apparent fraction or integer.
More generally, given a reducible fraction (the numerator and denominator share common factors other than unity), it can always be reduced (i.e. simplified) until obtaining an irreducible equivalent fraction. The notion of an irreducible fraction is generalized to the field of quotients of any single factorization domain: every element of this field can be written as a fraction in which the numerator and denominator are coprime.
Comparison of fractions
Comparing two fractions is used to check which is larger. There are several cases, depending on the numerators and denominators of these. Fractions are said to be homogeneous if they have the same denominator and fractions are heterogeneous if they have different denominators.
If the fractions are homogeneous—the denominator of the two fractions is the same—the fraction with the largest numerator is greater than the other.
If the numerator of the two positive fractions is the same, the fraction with the smaller denominator is greater than the other. This is quite natural: if you have two identical cakes, one to be distributed among more people than the other, the one that is distributed among fewer people will be divided into larger portions.
One way to compare fractions with different numerators and denominators is to find a common denominator. To compare and, convert and into equivalent fractions. Then bd is a common denominator and the numerators ad and bc can be compared.
It is not necessary to determine the value of the common denominator to be compared. This shortcut is known as "cross multiplication." Only ad and bc are compared, without calculating the denominator.
By multiplying both parts of each fraction by the denominator of the other, a common denominator is obtained:.
The denominators are now equal, but it is not necessary to calculate their value – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), .
Generally, when you have to calculate the common denominator of fractions, the least common multiple (lcm) of the denominators of the original fractions is used, rather than the least common denominator of these.
Fractions can be greater, less or equal if they are compared with the unit.[7] To compare them, we proceed to compare the numerator and denominator of the fraction.
Addition and subtraction of fractions
To add or subtract fractions, two cases are distinguished: if they have the same denominator, then the numerators are added or subtracted and the common denominator is left.
It is possible that the result can be simplified:.
Or if they have different denominators, you have to obtain fractions equivalent to the given fractions, so that they have a common denominator and then add or subtract. For example.
This method can be expressed algebraically as.
In reality, it is not necessary to obtain equivalent fractions so that the resulting denominator is the product of the denominators of the initial fractions. Simply take the least common multiple of the denominators. At the end of the operation, another simplification may be necessary. This would be represented like this:
The least common multiple of the denominators (21) is taken and this is divided by the same denominators and then multiplied by their corresponding numerators. Thus, the numerator of the result is defined (21:7=3, 3*2=6 and 21:3=7, 7*1=7. Then 7+6=13). This is just a simplified form of the previous one.
Multiplication and division of fractions
To multiply two fractions, simply multiply the numerators on the one hand and the denominators on the other. As an example,
During the operation, if the numerator of one fraction and the denominator of another—and vice versa—have some common factor, it can be canceled, since it is multiplying and dividing by said factor in the resulting fraction. This shortcut is known as "cancellation" and allows you to reduce the terms to be multiplied. The general algebraic expression would be:
En la división de fracciones, el numerador de la fracción resultante es el producto del numerador de la fracción dividendo por el denominador de la fracción divisor, mientras que el denominador es igual al denominador de la fracción dividendo multiplicado por el numerador de la fracción divisor. Otra manera de imaginarlo es que dividir entre un número es lo mismo que multiplicar por el inverso de ese número, por lo que la división de dos fracciones es igual a la multiplicación de la primera fracción por el inverso de la segunda:.