Problem 35 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{190}{495}\)

Step-by-Step Solution

Answer

The fraction \( \frac{190}{495} \) as a repeating decimal is \( 0.\overline{38} \).

Step 1: Simplify the Fraction

First, let's simplify the fraction \( \frac{190}{495} \). To do this, we need to find the greatest common divisor (GCD) of 190 and 495. The GCD of 190 and 495 is 5. Thus, we divide both the numerator and the denominator by 5:\[ \frac{190}{495} = \frac{190 \div 5}{495 \div 5} = \frac{38}{99} \] Now, our simplified fraction is \( \frac{38}{99} \).

Step 2: Convert Fraction to Decimal

Now that we have the simplified fraction \( \frac{38}{99} \), our task is to convert it into a decimal. To convert \( \frac{38}{99} \) into a decimal, divide 38 by 99. When you carry out the division (38 ÷ 99), the decimal expansion starts repeating immediately after the decimal point.

Step 3: Identify the Repeating Pattern

Perform the long division: 38 divided by 99 gives a quotient of 0.3838383838... and so on.The digits '38' repeat indefinitely, so the repeating decimal pattern is '38'.The repeating decimal can be expressed using bar notation as \( 0.\overline{38} \).

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is an essential concept when simplifying fractions. It is the largest number that can divide two or more numbers without leaving a remainder. Imagine it as the biggest building block both numbers share.
To find the GCD of 190 and 495 in our problem, we look for numbers that can divide both 190 and 495 evenly.

• List the factors of 190: 1, 2, 5, 10, 19, 38, 95, 190
• List the factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495

By examining these factors, we can see that the highest number common to both lists is 5. Hence, 5 is the GCD of 190 and 495.
This means that both terms in the fraction can be divided evenly by 5, which helps simplify it for further calculations.

Fraction Simplification Process

Fraction simplification involves reducing a fraction to its simplest form. This means finding an equivalent fraction with the smallest possible numerator and denominator, which is done by dividing both parts by their GCD.
For example, starting with \(\frac{190}{495}\):- We use the GCD, which is 5, to divide both the numerator (190) and denominator (495).- After dividing, we get the simplified fraction \(\frac{38}{99}\).The reason for this simplification is to make our calculations easier and the numbers more manageable.
Simplified fractions are easier to interpret and convert into decimals directly. It's like cleaning up a math problem to see it more clearly!

Performing Long Division

Long division is a method to divide numbers to get a quotient and, sometimes, a remainder. For fractions like \(\frac{38}{99}\), we use long division to convert it into a repeating decimal.Here’s how it’s done:- Begin by dividing the numerator (38) by the denominator (99).- Write 38 under the division bar and divide it by 99, which fits zero times. Place a zero before the decimal point.- Bring down digits and continue the division to several decimal places.As seen from the division, 38 divided by 99 results in a repeating decimal: 0.383838... Notice that '38' repeats over and over.
To clearly show this repeating pattern, we use bar notation: \(0.\overline{38}\). The bar over '38' indicates this part of the decimal repeats endlessly. Long division helps us reveal these repeating sequences in decimals.