Problem 58 - Algebra A Combined Function

Question

Write each fraction as a decimal. If the decimal is a repeating decimal, write using the bar notation and then round to the nearest hundredth. $$ \frac{34}{9} $$

Step-by-Step Solution

Answer

The decimal is 3.7777... (3.\overline{7}), rounded to 3.78.

Step 1: Divide the Numerator by the Denominator

Begin by dividing 34 (numerator) by 9 (denominator) to convert the fraction into a decimal. Perform the division: \[ 34 \div 9 = 3.7777 ext{...} \] The result is a repeating decimal.

Step 2: Identify the Repeating Part

Observe the decimal obtained from the division. In the decimal \(3.7777\ldots\), the digit '7' is repeating. We express this using bar notation as \(3.\overline{7}\).

Step 3: Round to the Nearest Hundredth

To round \(3.\overline{7}\) to the nearest hundredth, consider the digits following the second decimal place. Here, we have a repeating '7'. Since any digit 5 or greater rounds the preceding digit up, round \(3.7777\ldots\) to \(3.78\).

Key Concepts

Repeating Decimal

When you convert a fraction to a decimal, you might encounter a repeating decimal. This happens when a pattern of one or more digits repeats indefinitely. Imagine you are dividing numbers and the remainder never reaches zero. Instead, you keep seeing the same remainder over and over.
For the fraction \( \frac{34}{9} \), after performing the division, the result 3.7777... appears. This means the digit '7' repeats without stop. Repeating decimals can make decimals look endless. Recognizing this pattern is important as it signals that the division process creates a repeating sequence.
Repeating decimals occur because not all numbers can be perfectly divided into each other giving an exact decimal without remainder.

Bar Notation

To simplify the representation of repeating decimals, we use bar notation. Bar notation places a horizontal line or 'bar' over the repeating digits of a decimal. This helps us avoid writing endless numbers and still clearly show the repeating part.
For example, instead of writing 3.7777... for \( \frac{34}{9} \), we write it more neatly as \( 3.\overline{7} \). This tells anyone reading the number that '7' repeats endlessly.
Using bar notation provides several benefits:

• It saves time and reduces errors in writing.
• It's a universal way to show repeating decimals.
• It makes working with these numbers in further calculations more manageable.

Rounding Decimals

Rounding decimals is a way to simplify numbers, making them easier to use in everyday calculations. Often, precise repetition like \( 3.\overline{7} \) is not necessary, and we can round to a specific decimal place, such as the nearest hundredth.
To round a repeating decimal like \( 3.7777... \), we consider the third digit in the sequence because we aim to round to two decimal places. In this case, the next digit after 3.77 is another '7'.
Since this digit (7) is 5 or higher, it means that we round the last '7' up, resulting in 3.78.
Rounding helps in:

• Making numbers more practical for financial transactions.
• Handling numbers efficiently when exact precision isn't necessary.
• Simplifying data input into calculators or computers.