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Chapter 5: Problem 151

In the following exercises, divide. $$0.6 \div 0.2$$

Short Answer

Expert verified
The result of dividing 0.6 by 0.2 is 3.

Step by step solution

01

Understand the Problem

The problem requires dividing 0.6 by 0.2. This needs to be solved using division.
02

Convert to Fractions

Convert the decimal numbers to fractions. \(0.6 = \frac{6}{10}\) and \(0.2 = \frac{2}{10}\)
03

Simplify the Division

Rewrite the division of fractions problem: \[ \frac{6}{10} \div \frac{2}{10} \] This is equivalent to \[ \frac{6}{10} \times \frac{10}{2} \], as division by a fraction is equivalent to multiplication by its reciprocal.
04

Perform the Multiplication

Now multiply the fractions: \[ \frac{6}{10} \times \frac{10}{2} = \frac{6 \times 10}{10 \times 2} = \frac{60}{20} \]
05

Simplify the Result

Simplify the fraction: \[ \frac{60}{20} = 3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Conversion
When dealing with decimal division, the first step is often to convert the decimals into fractions. This makes the calculations easier to handle.
In our example, we start with the decimal numbers 0.6 and 0.2. To convert these into fractions, we recognize the place value of each decimal.
For 0.6, the digit 6 is in the tenths place, which means it can be represented as \(\frac{6}{10}\). Similarly, for 0.2, the digit 2 is in the tenths place, so it can be represented as \(\frac{2}{10}\).
This conversion is crucial as it transforms the problem into a fraction division task, which can follow standard arithmetic rules.
Reciprocal
Understanding the reciprocal is key when dividing fractions. The reciprocal of a fraction is simply flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{2}{10}\) is \(\frac{10}{2}\). When you divide by a fraction, you actually multiply by its reciprocal.
In the given exercise, we convert the decimal division \(0.6 \div 0.2\) to fraction division \(\frac{6}{10} \div \frac{2}{10}\). To divide by \(\frac{2}{10}\), we multiply by its reciprocal, \(\frac{10}{2}\).
This changes the division problem into a multiplication problem, making it easier to solve.
Simplification
Simplifying fractions is the final step to make the result easier to understand and use.
After converting and multiplying, you may end up with a fraction that needs simplification. In our example, we multiplied \(\frac{6}{10} \times \frac{10}{2} \)to get \(\frac{60}{20}\).
Simplifying this fraction means dividing both the numerator and the denominator by their greatest common divisor (GCD). For \(\frac{60}{20}\), the GCD is 20.
So, \(\frac{60}{20} = 3\). Simplifying allows us to present our answer in its simplest form, making it more comprehensible and clean.

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