Simplifying Fractions (B)

The simplest form of a fraction is a fraction that cannot be reduced. That means, in the language of mathematics, that the numerator and denominator do not have any common factors.


How do we know if a fraction is in its simplest form? If a fraction is not in its simplest form, how do we turn it into its simplest form?

I. Definitions and property

A. Definitions

Definition 1: a and b are two whole numbers, and b ≠ 0; we say that we can simplify the fraction $\frac{a}{b}$ if a and b have a common factor k greater than or equal to 2.


In this case, there are two natural numbers c and d such that a = c × k and b = d × k and we can write: $\frac{a}{b}=\frac{{\displaystyle c\times k}}{{\displaystyle d\times k}}=\frac{c}{d}$; we say that we have simplified the fraction by k.


For example, the fraction $\frac{8}{6}$ can be simplified by 2, so that: $\frac{8}{6}=\frac{{\displaystyle 4\times 2}}{{\displaystyle 3\times 2}}=\frac{4}{3}$.


Definition 2: a and b are two whole numbers, and b ≠ 0. We say that the fraction $\frac{a}{b}$ is in its simplest form, or cannot be reduced, if it cannot be simplified any more.

B. Property

To say that a fraction $\frac{a}{b}$ is in its simplest form or that it cannot be reduced, means that a and b do not have any common factors apart from 1, or that the Greatest Common Factor (abbreviated GCF; sometimes also referred to as the Greatest Common Divisor or GCD) is 1. This can be stated mathematically as GCF(a, b) = 1.


For example, the fraction $\frac{7}{6}$ cannot be reduced because the only common factor of 7 and 6 is 1.


Using this property, to find out if a fraction $\frac{a}{b}$ is in its simplest form, we just need to calculate the GCF of a and b.
There are two possibilities:
if GCF(a, b) = 1, the fraction is in its simplest form;
if GCF (a, b)  ≠ 1, we can turn the fraction into its simplest form by simplifying by GCF(a, b).

II. Calculating the GCF of two whole numbers

A. The difference method

Property: If a and b are two non-zero whole numbers such that a > b, then: GCF(a, b) = GCF (b, a – b).


Method: To calculate the GCF of two whole numbers, we can apply this property several times, and, at each stage, we obtain smaller whole numbers. The two examples below demonstrate how the process ends.


Example 1: Calculate the GCF of 12 and 18. We have, successively: GCF(18, 12) = GCF(12, 6) = GCF(6, 6) = 6.


Example 2: Calculate the GCF of 45 and 32. We have, successively: GCF(45, 32) = GCF(32, 13) = GCF(19, 13) = GCF(13, 6) = GCF(7, 6) = GCF(6, 1) = 1.


The final stage of the process will be one of these:
GCF (n, n), where n ≠ 1, and, in this case, the GCF will be n;
GCF (n, 1), where n ≠ 1, and, in this case, the GCF will be 1.

B. Euclidean Algorithm

Property: If a and b are two non-zero whole numbers such that a > b and if r denotes the remainder in the division of a by b, then: GCF(a, b) = GCF(b, r). This property is called Euclidean algorithm because it uses this type of division with a remainder, which is sometimes called Euclidean division.


Method: To calculate the GCF of two natural whole numbers, we apply this property several times and we stop at the first division where the remainder is equal to zero. The GCF is then the last non-zero remainder of the series of divisions carried out. It is easiest to present the results in a table, as demonstrated in the following example.


Example: Calculate the GCF of 128 and 58.

The GCF of 128 and 58 is therefore equal to 2.

III. Examples of Simplifying Fractions

A. Recognizing a fraction that is in its simplest form

Example: We want to demonstrate that the fraction  cannot be reduced.


We calculate the GCF of 352 and 159, using the Euclidean algorithm. We obtain the following table.

Therefore we have: GCF(352, 159) = 1, which proves that the fraction $\frac{159}{352}$ is in its simplest form.

B. Transforming a fraction into its simplest form

Example: We want to turn the fraction $\frac{1,612}{1,519}$ into its simplest form.


We calculate the GCF of 1,612 and 1,519 using the Euclidean algorithm. We obtain the following table.

We have: GCF(1,612, 1,519) = 31. Therefore we can simplify the fraction by 31 and the fraction we obtain is in its simplest form: $\frac{1,612}{1,519}=\frac{{\displaystyle 31\times 59}}{{\displaystyle 31\times 49}}=\frac{52}{49}$.