Problem 33 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{98}{66}\)

Step-by-Step Solution

Answer

The fraction \( \frac{98}{66} \) as a repeating decimal is \(1.\overline{48} \).

Step 1: Simplify the Fraction

Start by simplifying the fraction. Find the greatest common divisor (GCD) of the numerator (98) and the denominator (66). The GCD is 2. Divide both the numerator and the denominator by their GCD: \[ \frac{98}{66} = \frac{98 \div 2}{66 \div 2} = \frac{49}{33} \]. Now the fraction is simplified to \( \frac{49}{33} \).

Step 2: Divide the Numerator by the Denominator

Now, convert the simplified fraction \( \frac{49}{33} \) into a decimal by dividing 49 by 33. First, divide 49 by 33 which goes once (1). Subtracting gives a remainder of 16.

Step 3: Continue the Division to Find the Repeating Pattern

Next, drop a zero to get 160. Divide 160 by 33, which goes 4 times (4), giving a product of 132. Subtract 132 from 160 to get a remainder of 28. Then, bring down another zero to get 280. Dividing 280 by 33 goes 8 times (8), giving 264. Subtract 264 from 280, resulting in 16—where the pattern (16) repeats.

Step 4: Identify and Notate the Repeating Decimal

Since after these steps we are back to a remainder of 16, this signifies the beginning of a repeating sequence. Thus, the recurring decimal part is \(1.484848\ldots\). Hence, using the repeating bar notation, the repeating decimal form of \( \frac{98}{66} \) is \(1.\overline{48} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is the first and often crucial step towards converting them into decimals or managing them in other mathematical computations. The idea is to express the fraction in its simplest form so we can work with it more efficiently. When you simplify a fraction, you're determining whether the numerator and denominator have common factors and canceling them out to make life easier. Here's how you do it:

• Identify the greatest common divisor (GCD) of the fraction's numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is a simplified fraction.

In the given problem, we started with \( \frac{98}{66} \). The GCD of 98 and 66 is 2. By dividing both terms by 2, the fraction simplifies to \( \frac{49}{33} \). Now, this fraction is simpler to use for further steps like converting it into a decimal.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is key to simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Knowing how to find the GCD efficiently can save you time and help you deal with fractions easily.Here’s how you can find the GCD:

• List the factors of each number involved in the fraction.
• Identify the largest factor that these numbers share.
• This number is your GCD.

For example, with the numbers 98 and 66, the factors of 98 include 1, 2, 7, 14, 49, and 98. The factors of 66 are 1, 2, 3, 6, 11, 22, 33, and 66. The biggest common factor is 2, making 2 the GCD. This allows us to simplify \( \frac{98}{66} \) to \( \frac{49}{33} \). Properly simplifying fractions with the help of the GCD makes calculations much smoother.

Long Division

Long division is a proven method to convert a simplified fraction into its decimal form, especially when dealing with repeating decimals. It's about dividing the numerator by the denominator step by step, tracking how many times the divisor fits into parts of the dividend.Here's a breakdown of the long division process:

• Start with your dividend (numerator) and divide it by your divisor (denominator) to find how many times it fits; write that number down.
• Multiply the divisor by the quotient you obtained, and subtract that from your original dividend to find the remainder.
• Bring down the next digit from the dividend, treating it as the new dividend, and repeat the process.

In our exercise, we used long division on \( \frac{49}{33} \). Initially, 49 divided by 33 goes once. After subtracting 33 from 49, the remainder is 16. Bringing down another zero results in 160. Proceed by dividing 160 by 33, which goes four times with a remainder. Bringing each zero down as necessary and keeping track of remainders will eventually highlight a repeating sequence of 48 in this case, resulting in the decimal representation \(1.\overline{48}\). This process not only turns fractions into decimals but also helps us recognize repeating patterns accurately.