Problem 31 - Algebra A Combined Function

Question

Find each reciprocal. 0.2

Step-by-Step Solution

Answer

The reciprocal of 0.2 is 5.

Step 1: Understand the concept of reciprocal

The reciprocal of a number is 1 divided by that number. For a decimal or fraction, the reciprocal simply inverses the numerator and denominator.

Step 2: Convert the decimal to a fraction

Convert 0.2 into a fraction. Since 0.2 has one digit after the decimal, it can be expressed as \(\frac{2}{10}\). Simplify \(\frac{2}{10}\) to \(\frac{1}{5}\) by dividing both the numerator and the denominator by 2.

Step 3: Find the reciprocal of the fraction

To find the reciprocal of \(\frac{1}{5}\), flip the fraction. The reciprocal of \(\frac{1}{5}\) is \(\frac{5}{1}\).

Step 4: Simplify if necessary

Since \(\frac{5}{1}\) is already simplified, you can write it as the whole number 5.

Key Concepts

Understanding Decimal to Fraction Conversion

Sometimes numbers can be a bit tricky when they’re in decimal form, but turning them into fractions simplifies them a lot. Imagine the decimal 0.2. To express 0.2 as a fraction, consider the placement of the decimal. Each digit after the decimal moves the number to a higher place value in the fraction (tenths, hundredths, etc.). With the example of 0.2, there is one digit after the decimal. This means that 0.2 can be written as \( \frac{2}{10} \).

Converting a decimal to a fraction involves placing the decimal part over the appropriate power of 10. So, for any decimal number 0.x, you place x over 10, making it \( \frac{x}{10} \). If it were 0.24, it would involve two digits, turning it into \( \frac{24}{100} \). Recognizing the decimal's position helps achieve an immediate conversion to a fraction. This is crucial for the next steps that involve simplifying and finding reciprocals.

Simplifying Fractions Basics

Fractions can often be simplified, meaning they can be reduced to their smallest form for easier comparison and calculation. Simplifying \( \frac{2}{10} \) turns it into \( \frac{1}{5} \). This involves finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing both by this number.

In our example, the number 2 is the GCD of 2 and 10. Doing the division, both 2 and 10 by 2, simplifies the fraction to \( \frac{1}{5} \). This process reduces fractions to their most basic form, often leading to clearer solutions and making further calculations easier.

When you simplify fractions, you're eliminating any common factors to reveal the most reduced form, but remember: the value of the fraction doesn't change. This practice is essential when dealing with reciprocal calculations.

Learning about Fraction Inversion

Inverting a fraction is a key concept because it helps find the reciprocal. The reciprocal of a number flips the roles of the numerator and denominator. For the fraction \( \frac{1}{5} \), its reciprocal is \( \frac{5}{1} \). You essentially exchange places, making what was on top go to the bottom and vice versa.

This concept is straightforward but very powerful in mathematics, especially when solving equations or during certain operations like division. Once you have the reciprocal, you need to check if the fraction can be simplified again. A simplified reciprocal, like \( \frac{5}{1} \), is often expressed as a whole number if possible, so it becomes 5.

Understanding fraction inversion enables efficient problem-solving, streamlining processes that involve division and multiplication in fraction form.