Question

Express each of the following fractions in their simplest form: (a) \(\frac{12}{60}\) (c) \(\frac{27}{81}\) (d) \(\frac{6}{92}\) (e) \(\frac{377}{390}\)

Short answer

Answer: The simplified fractions are (a) \(\frac{1}{5}\), (c) \(\frac{1}{3}\), (d) \(\frac{3}{46}\), and (e) \(\frac{29}{30}\).

Step-by-step solution

(Step 1: Finding the GCD for each fraction)

To find the greatest common divisor (GCD) for each fraction, we look for the largest number that can divide both the numerator and denominator. For example, the GCD of 12 and 60 is 12.

(Step 2: Simplifying the fractions using the GCD)

Once we find the GCD, we divide both the numerator and the denominator by the GCD, and then write the new fraction. For example, \(\frac{12}{60}\) simplifies to \(\frac{12\div 12}{60\div 12}\), which gives \(\frac{1}{5}\). Now, let's simplify the given fractions: (a) \(\frac{12}{60}\): The GCD of 12 and 60 is 12. Divide both numerator and denominator by 12 to simplify the fraction. \(\frac{12}{60} = \frac{12\div 12}{60\div 12} = \frac{1}{5}\) (c) \(\frac{27}{81}\): The GCD of 27 and 81 is 27. Divide both numerator and denominator by 27 to simplify the fraction. \(\frac{27}{81} = \frac{27\div 27}{81\div 27} = \frac{1}{3}\) (d) \(\frac{6}{92}\): The GCD of 6 and 92 is 2. Divide both numerator and denominator by 2 to simplify the fraction. \(\frac{6}{92} = \frac{6\div 2}{92\div 2} = \frac{3}{46}\) (e) \(\frac{377}{390}\): The GCD of 377 and 390 is 13. Divide both numerator and denominator by 13 to simplify the fraction. \(\frac{377}{390} = \frac{377\div 13}{390\div 13} = \frac{29}{30}\) So, the simplified fractions are: (a) \(\frac{1}{5}\) (c) \(\frac{1}{3}\) (d) \(\frac{3}{46}\) (e) \(\frac{29}{30}\)

Key concepts

Greatest Common Divisor

Understanding the Greatest Common Divisor (GCD), which is also known as the Greatest Common Factor (GCF), is crucial when simplifying fractions. The GCD of two numbers is the largest number that can evenly divide both of them without leaving a remainder. For instance, to find the GCD of 27 and 81, list the factors of each number.

• Factors of 27: 1, 3, 9, 27
• Factors of 81: 1, 3, 9, 27, 81

Here, the greatest factor they both share is 27, which is the GCD.
When simplifying a fraction, finding the GCD is the first step because it leads to the most reduced form of a fraction. Dividing both the numerator and the denominator by the GCD ensures that the fraction is in its simplest form, which means you cannot simplify it any further using elementary arithmetic.

Numerator and Denominator

Fractions are composed of two main parts: the numerator, which is the top number, and the denominator, which is the bottom number. The numerator represents how many parts you have, while the denominator represents how many parts the whole is divided into.
For example, in the fraction \(\frac{6}{92}\), 6 is the numerator and 92 is the denominator. When simplifying fractions, you focus on reducing both numbers while maintaining the same value of the whole fraction. It's like saying, instead of having 6 slices out of 92 slices of a huge pizza, we can also express it as 3 slices out of 46 slices of the same pizza if we cut each slice in half.

Elementary Arithmetic

Elementary arithmetic involves basic operations such as addition, subtraction, multiplication, and division, which are the foundation for manipulating fractions. When simplifying fractions, division is used to reduce the numerator and the denominator by their GCD. Take the fraction \(\frac{377}{390}\), for example. By dividing both numbers by their GCD, which is 13, we use division to simplify: \(\frac{377}{390} = \frac{377\div 13}{390\div 13} = \frac{29}{30}\). Elementary arithmetic is essential for fraction reduction, which relies on these basic operations to simplify to the lowest terms.

Fraction Reduction

Fraction reduction, also known as simplifying fractions, is the process of making a fraction as simple as possible. To reduce a fraction, you divide the numerator and the denominator by their greatest common divisor. This process ensures that the fraction represents the same value in its simplest form, where both the numerator and the denominator are as small as possible while still keeping the same ratio.
For instance, in the exercise \(\frac{27}{81}\), by dividing both the numerator (27) and the denominator (81) by their GCD (27), the fraction reduces to \(\frac{1}{3}\). Reduced fractions are easier to understand and work with, especially when dealing with adding, subtracting, multiplying, or dividing multiple fractions in a problem.