Question

Express each fraction in its simplest form: \(\begin{array}{lllll}\text { (a) } \frac{9}{12} & \text { (b) } \frac{16}{20} & \text { (c) } \frac{20}{16} \text { (d) } \frac{72}{96} & \text { (e) }-\frac{30}{42} \text { (f) } \frac{-20}{-45}\end{array}\)

Short answer

Question: Simplify the following fractions and express them in their simplest form: (a) \(\frac{9}{12}\) (b) \(\frac{16}{20}\) (c) \(\frac{20}{16}\) (d) \(\frac{72}{96}\) (e) \(-\frac{30}{42}\) (f) \(\frac{-20}{-45}\) Answer: (a) \(\frac{3}{4}\) (b) \(\frac{4}{5}\) (c) \(\frac{5}{4}\) (d) \(\frac{3}{4}\) (e) \(-\frac{5}{7}\) (f) \(\frac{4}{9}\)

Step-by-step solution

(a) Simplifying \(\frac{9}{12}\)

First, find the GCD of 9 and 12. The GCD is 3. Now, divide both the numerator and the denominator by 3: \(\frac{9}{12} = \frac{9\div3}{12\div3} = \frac{3}{4}\)

(b) Simplifying \(\frac{16}{20}\)

Find the GCD of 16 and 20. The GCD is 4. Divide both the numerator and the denominator by 4: \(\frac{16}{20} = \frac{16\div4}{20\div4} = \frac{4}{5}\)

(c) Simplifying \(\frac{20}{16}\)

Find the GCD of 20 and 16. The GCD is 4. Divide both the numerator and the denominator by 4: \(\frac{20}{16} = \frac{20\div4}{16\div4} = \frac{5}{4}\)

(d) Simplifying \(\frac{72}{96}\)

Find the GCD of 72 and 96. The GCD is 24. Divide both the numerator and the denominator by 24: \(\frac{72}{96} = \frac{72\div24}{96\div24} = \frac{3}{4}\)

(e) Simplifying \(-\frac{30}{42}\)

Find the GCD of 30 and 42. The GCD is 6. Divide both the numerator and the denominator by 6 and keep the negative sign: \(-\frac{30}{42} = -\frac{30\div6}{42\div6} = -\frac{5}{7}\)

(f) Simplifying \(\frac{-20}{-45}\)

Find the GCD of 20 and 45. The GCD is 5. Divide both the numerator and the denominator by 5. The negative signs cancel out, resulting in a positive fraction: \(\frac{-20}{-45} = \frac{-20\div5}{-45\div5} = \frac{4}{9}\)

Key concepts

Greatest Common Divisor (GCD)

Understanding the Greatest Common Divisor (GCD) is essential for simplifying fractions effectively. The GCD, also known as the greatest common factor, is the highest number that divides exactly into two or more numbers without leaving a remainder. In the context of fractions, the GCD refers to the largest number that can divide both the numerator (the top number) and the denominator (the bottom number).

For instance, to find the GCD of 9 and 12, we list the factors of each: the factors of 9 are 1, 3, and 9, while the factors of 12 are 1, 2, 3, 4, 6, and 12. The common factors are 1 and 3, and the greatest of these is 3. Hence, the GCD of 9 and 12 is 3. Knowing the GCD allows you to reduce a fraction to its simplest form by dividing both terms by this common value.

Numerators and Denominators

When dealing with fractions, it's essential to know the roles of numerators and denominators. The numerator, located above the fraction bar, represents the number of parts we have. The denominator, below the fraction bar, signifies the total number of equal parts that something is divided into. A fraction thus expresses the ratio of the numerator to the denominator.

In the fractions \(\frac{9}{12}\) or \(\frac{16}{20}\), 9 and 16 are numerators indicating 'how many' we have, while 12 and 20 are denominators telling us 'out of how many.' To simplify fractions, we look for a GCD to reduce the numerator and denominator to smaller numbers that maintain the same value of the fraction.

Simplest Form of a Fraction

The simplest form of a fraction, also known as its lowest terms, is when the numerator and denominator are as small as possible while still keeping the same value of the original fraction. This means that there is no number, other than 1, that divides both the numerator and denominator exactly.

For example, the fraction \(\frac{9}{12}\) can be simplified to \(\frac{3}{4}\) after dividing both the numerator and the denominator by their GCD, which is 3. After the reduction, there is no number other than 1 that can evenly divide both 3 and 4, hence \(\frac{3}{4}\) is in its simplest form. Ensuring that a fraction is in its simplest form is crucial for clarity in mathematical expressions and operations.

Reducing Fractions

Reducing fractions is the process of converting them to the simplest form where the numerator and denominator have no common factors other than one. This makes calculations easier and results more understandable.

A step-by-step guide to reducing fractions is as follows: First, determine the GCD of the numerator and denominator. Then, divide both the numerator and denominator by the GCD. Keep the negative sign in its original position if the fraction is negative, as seen in fraction \(\frac{-30}{42} \), where the fraction becomes \( -\frac{5}{7} \) after division by the GCD, which is 6. Remember that when a fraction has negative signs in both the numerator and denominator, they cancel each other out, and the fraction remains positive, as in \(\frac{-20}{-45} \) simplifying to \(\frac{4}{9}\).