Problem 80 - Intermediate Algebra

Question

Explain the difference between \(1 . \overline{3}\) and \(1.3\).

Step-by-Step Solution

Answer

\(1 . \overline{3} > 1.3\) since it equals \(\frac{4}{3}\) while \(1.3 = \frac{13}{10}\).

Step 1: Understand the Numbers

First, understand the notation of the numbers. The number \(1 . \overline{3}\) is a repeating decimal, meaning it is represented as \(1.3333\ldots\) indefinitely. The number \(1.3\) is a terminating decimal, with a clear end after the first decimal place.

Step 2: Convert Repeating Decimal to Fraction

Convert the repeating decimal \(1 . \overline{3}\) to a fraction. Let \(x = 1.3333\ldots\). Then \(10x = 13.3333\ldots\). Subtracting these gives \(9x = 12\), so \(x = \frac{12}{9} = \frac{4}{3}\). Therefore, \(1 . \overline{3} = \frac{4}{3}\).

Step 3: Convert Terminating Decimal to Fraction

Convert the terminating decimal \(1.3\) to a fraction. Since there is one digit after the decimal, \(1.3\) can be expressed as \(\frac{13}{10}\). Thus, \(1.3 = \frac{13}{10}\).

Step 4: Compare the Fractions

Now compare the fractions \(\frac{4}{3}\) and \(\frac{13}{10}\). Find a common denominator to compare: the least common multiple of 3 and 10 is 30. Convert \(\frac{4}{3}\) to \(\frac{40}{30}\) and \(\frac{13}{10}\) to \(\frac{39}{30}\). Notice that \(\frac{40}{30} > \frac{39}{30}\).

Key Concepts

Fraction Conversion

Fraction conversion is the process of changing decimals into fractions. Decimals can either terminate or repeat. When a decimal terminates, like \(1.3\), it stops after a certain number of digits. Converting a terminating decimal is simpler as you just place the decimal over a power of ten. For \(1.3\), since there's one digit after the decimal, it becomes \(\frac{13}{10}\).

Repeating decimals, like \(1 . \overline{3}\), go on forever with the same digits. To convert them into fractions, you must set up an equation. For example, let a repeating decimal be \(x = 1.3333\ldots\). Multiplying by 10 gives \(10x = 13.3333\ldots\). By subtracting, you cancel out the repeating parts, so \(9x = 12\). Solve for \(x\) to find it equals \(\frac{4}{3}\). This way, repeating decimals are expressed as fractions without endless digits.

Decimal Notation

Decimal notation is how we represent numbers using a base-ten system that includes a decimal point. It's common for everyday calculations and measurements. A key aspect to understand is whether the decimal is terminating or repeating. Terminating decimals, like \(1.3\), have a finite number of digits. Repeating decimals, such as \(1 . \overline{3}\), display a sequence of digits which repeats indefinitely.

Recognizing the type of decimal helps in evaluating and comparing numbers. The repeating notation \(1 . \overline{3}\) signifies an infinite string of threes after the decimal, while \(1.3\) ends right after the 3. This difference is crucial when performing conversions or comparisons. Grasping these notations enables efficient calculation and clear understanding of number values in real-world scenarios.

Comparing Fractions

Comparing fractions involves determining which of two fractions is greater. Once you've converted decimals to fractions, they can be compared more easily. You need a common denominator to directly see which is larger. The fractions \(\frac{4}{3}\) and \(\frac{13}{10}\) must share a common denominator to compare.

The least common denominator of 3 and 10 is 30. Convert \(\frac{4}{3}\) to \(\frac{40}{30}\) and \(\frac{13}{10}\) to \(\frac{39}{30}\). Now, it's clear that \(\frac{40}{30}\) is greater than \(\frac{39}{30}\). When fractions have the same denominator, compare numerators directly. This technique helps see which fraction represents a larger quantity, thereby clarifying how decimals compare when converted into fractions.