Problem 42 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{686}{231}\)

Step-by-Step Solution

Answer

\( \frac{686}{231} = 2.\overline{96} \).

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{686}{231} \). To do this, find the greatest common divisor (GCD) of 686 and 231. The GCD is 7. Divide both numerator and denominator by 7 to get \( \frac{98}{33} \).

Step 2: Divide the Numerator by the Denominator

Now, divide 98 by 33 to convert the fraction into a decimal. Perform the long division of 98 by 33. The remainder repeats every 3 steps: 98 divided by 33 gives a quotient of 2 with a remainder of 32, then continue with 320, and so on.

Step 3: Identify the Repeating Cycle

Observe the decimal output from the long division: 2.969696... Notice that the digits '96' repeat endlessly. Hence, we can conclude that this is a repeating decimal.

Step 4: Use Repeating Bar Notation

Express the repeating decimal in repeating bar notation. The repeating sequence '96' should have a bar over it. Thus, \( \frac{686}{231} = 2.\overline{96} \).

Key Concepts

Fraction Simplification

Fraction simplification is an essential process in mathematics. It makes fractions easier to work with by reducing them to their simplest form. You can achieve this by finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both by that number. This process doesn't change the value of the fraction, just how it looks.
In our example with the fraction \(\frac{686}{231}\), we first need to find the GCD. The GCD of 686 and 231 is 7. By dividing both the numerator and the denominator by 7, we simplify the fraction to \(\frac{98}{33}\).
Remember that a completely simplified fraction will have a GCD of 1, meaning no further reduction is possible. Simplifying fractions not only makes them cleaner but also easier to work with in calculations.

Long Division

Long division is a technique used to divide one number by another, resulting in a quotient and often a remainder. It is particularly useful when dealing with large numbers or when converting fractions to decimals.
In our example, to convert \( \frac{98}{33} \) into a decimal, we perform long division. You start by dividing the first few digits of the dividend (the number being divided), here being 98, by the divisor, which is 33. Here, 33 goes into 98 two times, giving a remainder when you subtract the product of the divisor and quotient from the original number.

• Write down the initial remainder.
• Bring down the next digit from the dividend.
• Repeat the division process with the new number.

In this case, dividing 98 by 33 repeatedly shows that we reach a remainder that causes the sequence of digits to start repeating every 3 steps. This repeating process is what forms the repeating decimal 2.969696...
Long division helps identify these cycles of repetition, crucial for understanding repeating decimals.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest number that divides two or more given numbers without leaving a remainder. It is a key concept in simplifying fractions as it allows us to reduce them to their simplest form.
One way to find the GCD is to list out the factors of each number and find the largest one they share. Another way is the Euclidean algorithm, which is efficient for large numbers. To use it:

• Divide the larger number by the smaller one.
• Take the remainder and divide the smaller number by this remainder.
• Continue this process until a remainder of 0 is reached.
• The last non-zero remainder is the GCD.

For the fraction \(\frac{686}{231}\), using such methods, we find the GCD to be 7. This understanding of GCD not only aids in simplifying fractions but also plays a role in factoring polynomials and number theory comprehensions.