Question

Write each percent as a decimal number. $$6 \frac{2}{3} \%$$

Short answer

Answer: The decimal equivalent of $$6 \frac{2}{3} \%$$ is approximately $$0.06666$$ or $$0.067$$ (rounded to three decimal places).

Step-by-step solution

Convert the mixed number to an improper fraction

To convert $$6 \frac{2}{3}$$ into an improper fraction, we first multiply the whole number (6) by the denominator (3) and then add the numerator (2). $$\frac{6 \times 3 + 2}{3}$$ Now, calculate the result: $$\frac{18 + 2}{3} = \frac{20}{3}$$ The improper fraction is $$\frac{20}{3}$$.

Convert the improper fraction to decimal number

To convert the improper fraction $$\frac{20}{3}$$ into a decimal number, we divide the numerator (20) by the denominator (3): $$\frac{20}{3} = 6.666...$$ So the decimal equivalent of the fraction is 6.666...

Convert the percent to a decimal number

Now we have $$6.666... \%$$ (the equivalent decimal number of the given mixed number percent) and to convert it to a decimal number, we need to divide by 100: $$6.666... \div 100 = 0.06666....$$ Therefore, the decimal number equivalent to the given percent $$6 \frac{2}{3} \%$$ is $$0.06666....$$.

Key concepts

Understanding Improper Fractions

When dealing with fractions, there are two types that are often encountered: proper fractions and improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This might appear counterintuitive, since we usually think of fractions as numbers less than one. But improper fractions are just as valid and can represent quantities greater than or equal to one.

Improper fractions are useful in mathematical calculations because they facilitate certain operations, like addition and subtraction of fractions, without having to convert between mixed numbers and fractions. To convert a mixed number into an improper fraction, as in our exercise, you multiply the whole number by the denominator and add the numerator. The result is placed over the original denominator. For example, in the mixed number \(6 \frac{2}{3}\), you would do the following: \[\frac{6 \times 3 + 2}{3} = \frac{20}{3}\].

This conversion is important because it allows us to work with a single fraction rather than a combination of a whole number and a fraction, which simplifies subsequent steps, like converting to a decimal.

Decimal Conversion Process

Converting fractions to decimals is a fundamental skill in mathematics. This conversion is straightforward: you simply divide the numerator by the denominator. If the fraction is an improper one, the same rule applies, and you may end up with a decimal number greater than one. In our exercise example, the improper fraction \(\frac{20}{3}\) is converted to a decimal by dividing 20 by 3, yielding a decimal that repeats indefinitely: \[6.666...\].

These kinds of decimals, ones that go on forever without repeating a pattern, are called repeating decimals. The '...' indicates that the number continues infinitely. In practice, when using decimals, you usually round them to a certain number of decimal places for ease of calculation or presentation. Understanding this step of conversion is crucial to many fields, including finance, engineering, and science where precision is valued.

Converting Percent to Decimal

The relationship between percentages and decimals is integral to understanding various mathematical and real-world applications. Percent means per hundred, so when you are converting a percent to a decimal, you are essentially dividing it by 100. The process is simplified by moving the decimal point two places to the left. For instance, in our exercise we had \(6.666...\%\), and dividing by 100 we get \[0.06666...\].

It's good practice to remember that the percent sign is a shortcut for division by 100. Whether you are dealing with a whole number, a mixed number, or even an improper fraction as a percentage, the conversion principle remains the same. This concept is widely used in various aspects of life, including calculating discounts, interest rates, and statistical data. By mastering percent to decimal conversion, you empower yourself to tackle a broad array of mathematical problems and practical scenarios.