Problem 59 - Beginning and Intermediate Algebra

Question

Find each reciprocal or multiplicative inverse. $$ 0.2 $$

Step-by-Step Solution

Answer

The reciprocal of 0.2 is 5.

Step 1: Understanding the Concept of Reciprocal

The reciprocal (or multiplicative inverse) of a number is a value that, when multiplied with the original number, gives 1. For any number \(a\), its reciprocal is \(\frac{1}{a}\). In other words, if you multiply a number by its reciprocal, the result is always 1.

Step 2: Convert Decimal to Fraction

Given the decimal 0.2, first convert it to a fraction. The decimal 0.2 can be expressed as a fraction \(\frac{2}{10}\). Simplify the fraction by dividing the numerator and the denominator by 2 to get \(\frac{1}{5}\).

Step 3: Finding the Reciprocal

To find the reciprocal of \(\frac{1}{5}\), you need to flip the fraction. This means the reciprocal is \(\frac{5}{1}\), which simplifies to 5.

Key Concepts

Multiplicative Inverse

The multiplicative inverse, also known as the reciprocal, transforms a number in such a way that when you multiply the original number by this reciprocated value, the result is 1. If we represent a number by \( a \), its multiplicative inverse is \( \frac{1}{a} \). This property is essential in mathematics as it helps in defining division operations. Understanding the importance of the multiplicative inverse simplifies complex mathematical problems and aids in solving fractions.

When dealing with whole numbers or simple fractions, finding the multiplicative inverse involves flipping the fraction upside down. For example, the inverse of \( \frac{3}{4} \) would be \( \frac{4}{3} \). It's crucial to note that zero does not have a multiplicative inverse because any number multiplied by zero remains zero, never reaching 1.

Decimal to Fraction Conversion

Converting decimals to fractions is an important foundational skill in mathematics. This conversion is especially useful for operations involving fractions such as addition, subtraction, multiplication, and division. Let's demonstrate this using 0.2.

• First, recognize the place value of the decimal. In 0.2, the digit "2" is in the tenths position. Therefore, 0.2 can be written as \( \frac{2}{10} \).
• Next, simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. For \( \frac{2}{10} \), the GCD is 2.
• Divide both the numerator and the denominator by their GCD to get \( \frac{1}{5} \).

Understanding this process makes it easy to work with decimals and fractions interchangeably, enhancing problem-solving abilities in various mathematical contexts.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its smallest form while maintaining equal value. This is done by dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).

Let's take the fraction \( \frac{2}{10} \) as an example. Both 2 and 10 can be divided by 2, their GCD, to simplify the fraction to \( \frac{1}{5} \).
This process is fundamental because:

• It allows for easier manipulation of fractions. Performing tasks such as addition or subtraction becomes simpler when fractions are in their lowest terms.
• It is crucial in comparing fractions to determine which is larger or smaller.

Simplifying makes calculations more straightforward and avoids unnecessary complexity in mathematical equations.