Problem 92 - Algebra A Combined Function

Question

The notation \(0 . \overline{67}\) is the same as a. \(0.6777 \ldots\) b. \(0.67666 \ldots\) c. \(0.6767 \ldots\)

Step-by-Step Solution

Answer

c. \(0.6767 \ldots\)

Step 1: Understanding Repeating Decimals

The notation \(0.\overline{67}\) represents a repeating decimal where the digits 67 repeat indefinitely. This means \(0.\overline{67} = 0.67676767\ldots\).

Step 2: Comparing Decimals

We need to compare the repeating decimal \(0.676767\ldots\) to the given options. The correct match will have the same repeating pattern after the decimal point as \(0.67676767\ldots\).

Step 3: Analyzing Each Option

- Option a: \(0.6777\ldots\) suggests a repeating decimal of 7 only, starting with a 6. This doesn't match our pattern.- Option b: \(0.67666\ldots\) suggests a repeating 6 pattern, starting with 67. This is incorrect as well.- Option c: \(0.6767\ldots\) accurately reflects the repeating sequence of 67. This matches the pattern we need.

Step 4: Selecting the Correct Option

Given that option c, \(0.6767\ldots\), matches the repeating sequence after the decimal point, it correctly represents \(0.\overline{67}\).

Key Concepts

Decimal Notation

Decimal notation is a way of representing numbers that include fractions using a dot (also known as a decimal point). This notation is essential for understanding numbers that aren't whole.
For example, in the number 1.23, the digits after the decimal point (23) indicate the fractional part of the number. Each place after the decimal represents a power of ten less, such as tenths, hundredths, and so forth.
Repeating decimals have a sequence of digits that continuously repeat. In the notation \(0.\overline{67}\), it indicates that the digits '67' are repeated infinitely. This makes it different from terminating decimals where the fractions cease after a certain point.

To show a repeating sequence, we use a bar (called "vinculum") on top of the repeated digits.

This makes it easy to differentiate between numbers like \(0.67\) (where '67' ends) and \(0.\overline{67}\) (where '67' repeats forever).

Comparing Decimals

When comparing decimals, it’s crucial to pay attention to every detail, especially when dealing with repeating numbers. Comparing \(0.67676767...\) and \(0.6767...\) means we focus on their repeating patterns.
Matching repeating decimals means identifying exactly how the sequence repeats. You need to check the digits that appear after the decimal point to see if they follow a consistent and identical pattern.
For instance:

• Option a: \(0.6777...\) repeats '7' but starts with '6'. This is different since it lacks the continual '67' repetition.
• Option b: \(0.67666...\) indicates a sequence of '6' repeating, which begins with '67' but drops the consistent cycle.
• Option c: \(0.6767...\) perfectly mirrors our sequence of '67'. Therefore, it matches the original repeating decimal \(0.\overline{67}\).

Always ensure that both the start and the repeating sections are identical for an accurate match.

Understanding Patterns in Mathematics

Patterns in mathematics can help us predict and understand complex sequences. Repeating decimals are one such pattern. They display an endless cycle of digits, indicating precise repetition.
Recognizing these patterns requires a keen eye for consistency in sequences. Once you know the original sequence, identifying identical or differing sequences becomes clearer.

• Mathematical patterns can help simplify calculations by knowing what comes next without writing the entire sequence every time.
• In repeating decimals specifically, once you identify the repeating unit, you comprehend how it looks in any notation.
• This understanding aids in solving complex problems and ensures accuracy in computations.

Identifying and using patterns efficiently transforms how we handle problems, especially those involving intricate repetitions, like with decimals.