Express each fraction as an equivalent fraction with a denominator of 32 : (a) \(\frac{1}{16}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{4}\)

Understanding fraction conversion is essential when working with mathematics, especially when the goal is to find equivalent fractions with a common denominator.

Let's start with the basics: a fraction represents a part of a whole and consists of a numerator (the top number) and a denominator (the bottom number). The process of converting a fraction to an equivalent one involves finding a different fraction that represents the same amount but uses different numbers for the numerator and the denominator.

To make fractions equivalent, we look for a common denominator, which in the exercise example is 32. The key is to multiply both the numerator and the denominator of the original fraction by the same number. This number is essentially the factor that, when applied to the original denominator, will result in the common denominator. For instance, when converting \(\frac{1}{16}\) to have a denominator of 32, we multiply both the numerator and denominator by 2, resulting in \(\frac{2}{32}\), an equivalent fraction.

A crucial step in creating equivalent fractions involves the multiplication of both the numerator and the denominator by the same factor. This is what we call 'denominator multiplication' because it often starts with the goal of adjusting the denominator to a specific value.

Why do we multiply both parts of the fraction? It's important to maintain the fraction's value. Multiplying only the denominator would change the value of the fraction, which isn't our goal. Instead, by applying the same factor to the numerator, we ensure that the fraction's value remains the same.

In the provided exercise, the factors used are 2, 4, and 8 for the respective fractions \(\frac{1}{16}\), \(\frac{3}{8}\), and \(\frac{1}{4}\). This careful application of denominator multiplication achieves the target denominator of 32 without changing the value of the original fractions.

Numerical reasoning involves analyzing and interpreting numbers to make logical decisions. When it comes to equivalent fractions, numerical reasoning helps us understand why and how we convert fractions to a common denominator.

This type of reasoning comes into play when selecting the appropriate factor to multiply the denominators. One must consider the multiples of the current denominator and how they relate to the desired denominator. In our examples, numerical reasoning led us to multiply \(\frac{1}{16}\) by 2, \(\frac{3}{8}\) by 4, and \(\frac{1}{4}\) by 8. It's the skill that melds our understanding of numbers with practical application, ensuring that we accurately find equivalents without changing the original proportions of the fractions.