Problem 34 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{305}{330}\)

Step-by-Step Solution

Answer

The fraction \( \frac{305}{330} \) equals \(0.9\overline{24}\).

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{305}{330} \). Find the greatest common divisor (GCD) of the numerator (305) and the denominator (330). The GCD is 5, so divide both by 5: \( \frac{305 \div 5}{330 \div 5} = \frac{61}{66} \).

Step 2: Perform Long Division

Divide 61 by 66 using long division to find the decimal representation. Since 61 is less than 66, add a decimal point and continue the division. It gives \(0.92424\ldots\). The repeated sequence "24" starts after one decimal place.

Step 3: Identify the Repeating Part

Notice that the quotient 0.92424... shows that the digits "24" repeat indefinitely after the initial 9. So, the repeating decimal part is "24".

Step 4: Use Repeating Bar Notation

Express the repeating decimal using a bar notation. The non-repeating part is "9" and the repeating part is "24", so it is written as \(0.9\overline{24}\).

Key Concepts

Fraction Simplification

When working with fractions, one of the first things to do is to check if the fraction can be simplified. Simplifying a fraction means reducing it to its simplest form. This involves finding the common factors of the numerator and the denominator. By dividing both the top number (numerator) and the bottom number (denominator) by their greatest common divisor (GCD), you arrive at an equivalent fraction that is easier to work with.
For example, to simplify the fraction \(\frac{305}{330}\), we first find the GCD of 305 and 330, which is 5. By dividing both numbers by this GCD, we get \(\frac{61}{66}\). This fraction is in its simplest form, making any further calculations much cleaner and often less error-prone. Simplified fractions not only look nicer, but they also make comparisons easier and lay the groundwork for more complex calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept in fraction simplification. It refers to the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial because it helps simplify fractions.
Here are some simple steps to find the GCD of two numbers:

• List the factors for each number separately.
• Identify the common factors.
• Choose the largest factor that appears in both lists. This is your GCD.

A fast way to find the GCD is using the Euclidean algorithm, though for small numbers, simple factor listing is often the quickest for beginners. In our example with 305 and 330, we found the GCD to be 5, allowing us to simplify \(\frac{305}{330}\) to \(\frac{61}{66}\).
Understanding how to find and use the GCD is a powerful tool when simplifying fractions, ensuring you work with the most efficient and manageable numbers possible.

Long Division

Long division is a method used to divide larger numbers that cannot be divided easily in your head. It's especially important when handling non-integral answers, such as converting fractions to decimals. Let's break down the steps to see how it applies.
In this example, we'll use long division to convert \(\frac{61}{66}\) into a decimal:

• Begin by setting up the division, with 61 as the dividend and 66 as the divisor.
• Since 61 is less than 66, it's important to add a decimal point to the quotient.
• Proceed by adding zeros to 61 one at a time, making it 610, 6100, etc., to continue the division process.
• Work through division as you would normally, subtracting the largest multiple of 66 from the developed dividend number.
• Note the quotient pattern appearing. Here, you get \(0.92424\ldots\), showing the digits "24" repeat indefinitely.

By keeping track of repeating patterns, you can express the decimal succinctly using bar notation. For our example, the long division revealed a recurring number "24" after the initial "9", so the decimal is written as \(0.9\overline{24}\).
Long division not only helps in converting fractions to repeating decimals but also deepens your understanding of how fractional values translate into decimal form.