46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
47.0.3̅ = 0.333…

Verified step by step guidance

1. Identify the repeating decimal: here it is \(0.3\overline{3} = 0.333\ldots\) where the digit 3 repeats indefinitely.
2. Express the repeating decimal as an infinite geometric series. The first term is \(\frac{3}{10}\) (since 3 is in the tenths place), and each subsequent term is multiplied by \(\frac{1}{10}\) because the decimal repeats every place value. So the series is: \(\frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \cdots\).
3. Recognize this as a geometric series with first term \(a = \frac{3}{10}\) and common ratio \(r = \frac{1}{10}\).
4. Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(|r| < 1\).
5. Substitute \(a = \frac{3}{10}\) and \(r = \frac{1}{10}\) into the formula to write the decimal as a fraction: \(S = \frac{\frac{3}{10}}{1 - \frac{1}{10}}\).

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. For example, 0.3̅ means 0.333..., where the digit 3 repeats endlessly. Understanding this helps in converting such decimals into exact fractions.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Repeating decimals can be expressed as infinite geometric series, where each term represents a decimal place value.

Conversion of Infinite Geometric Series to Fractions

An infinite geometric series with first term 'a' and common ratio 'r' (|r|<1) converges to a/(1-r). This formula allows us to convert the geometric series representation of a repeating decimal into a fraction, providing an exact ratio of two integers.