Problem 24 - Elementary Algebra

Question

For exercises \(13-24\), rewrite the fraction as a decimal number. $$ \frac{2}{7} $$

Step-by-Step Solution

Answer

The decimal form of \(\frac{2}{7}\) is \(0.\bar{285714}\).

Step 1 - Understand the problem

The task is to convert the fraction \(\frac{2}{7}\) into a decimal number.

Step 2 - Divide the numerator by the denominator

Perform the division \(2 \div 7\). Using long division or a calculator will help to simplify this. The division gives a repeating decimal.

Step 3 - Write down the result

After performing the division, we get \(0.285714\) which repeats. Therefore, the decimal form of \(\frac{2}{7}\) is \(0.\bar{285714}\). The bar indicates that the sequence 285714 repeats indefinitely.

Key Concepts

fractions

A fraction represents a part of a whole. It is written as \(\frac{a}{b}\), where \( a \) is the numerator (the number of parts you have) and \( b \) is the denominator (the number of equal parts the whole is divided into). For example, in the fraction \(\frac{2}{7}\), 2 is the numerator and 7 is the denominator. Understanding fractions is important, as they are used in many aspects of everyday life, such as cooking or dividing resources.

decimals

A decimal number represents a fraction whose denominator is a power of ten. Converting fractions to decimals makes calculations easier and helps in comparing different values. For example, \(0.5\) means \(\frac{5}{10}\), which is equivalent to \(\frac{1}{2}\). Decimal numbers are key in many areas such as measurements, money, and scientific data.

repeating decimals

A repeating decimal is a decimal number that has digits that repeat infinitely. They are usually represented with a bar over the repeating part. For instance, after dividing 2 by 7, we get \(0.285714\) as the repeating sequence. Therefore, we write this as \(0.\overline{285714}\). Recognizing repeating decimals is crucial because it indicates that the fraction cannot be expressed as a finite decimal.

long division

Long division is a method for dividing two numbers to get a quotient and possibly a remainder. It involves dividing the numerator by the denominator step by step. For the fraction \(\frac{2}{7}\), we use long division to find the decimal form. The steps are as follows: Divide 2 by 7, which does not go completely, so we extend to 20 by adding decimal points and zeros. We then get 0.285714, which repeats. Mastering long division is helpful for converting fractions to decimals and solving many mathematical problems.