Problem 43 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{47}{66}\)

Step-by-Step Solution

Answer

The fraction \(\frac{47}{66}\) converts to the repeating decimal \(0.7\overline{12}\).

Step 1: Perform the Division

To convert the fraction \(\frac{47}{66}\) into a decimal, start by dividing 47 by 66. Since 66 goes into 47 zero times, we start by considering 470 (by placing a decimal and adding a zero) and determine how many times 66 fits into 470.

Step 2: Approximate the Quotient

Divide 470 by 66, which gives approximately 7. Place 7 in the decimal quotient after zero to get a partial result of 0.7. Subtract \(66 \times 7 = 462\) from 470 to get a remainder of 8.

Step 3: Continue the Division Process

Bring down a zero to make the remainder 80 and determine how many times 66 fit into 80, which is once (giving 66). Place 1 in the next decimal place to have 0.71, then subtract 66 from 80 to get 14.

Step 4: Explore the Repeating Cycle

Again, bring down a zero, making it 140. Divide 140 by 66 to get 2, giving \(132\) (since \(66 \times 2 = 132\)). Place 2 as the next digit in the decimal, then subtract 132 from 140 to get 8. Notice the cycle of remainders 8 to 80 to 140 repeats, confirming the repeating part.

Step 5: Write the Repeating Decimal

The decimal expansion of \(\frac{47}{66}\) is determined as the sequence 0.7121212... as the cycle continues with repeating digits 12. Thus, it is written as \(0.7\overline{12}\), indicating that 12 repeats indefinitely.

Key Concepts

Fractions to Decimals

Converting fractions to decimals can seem tricky at first, but it's a simple process of division. Every fraction is essentially a division problem where the numerator is divided by the denominator. In simplest terms, it involves seeing what part of the whole (denominator) is represented by the numerator. In our example, the fraction \( \frac{47}{66} \) means dividing 47 by 66. This division reveals the decimal form of the fraction. Decimal representation can be terminating (ending) or repeating. In repeating decimals, a specific sequence of digits pops up over and over indefinitely. This pattern can be easily identified and denoted using a repeating bar over the repeating digits.

Long Division

Long division is a method used to divide larger numbers and it is also useful for converting fractions to decimals. Here's how it works using the example of \( \frac{47}{66} \):

• First, determine how many times the divisor (66) goes into the initial number (47). Since 66 is larger, the initial result is 0.
• Next, consider the next whole number by adding a zero, making it 470. Now, divide this by 66.
• The approximate whole number quotient is 7, providing a partial result of 0.7.
• Subtract \(66 \times 7 = 462\) from 470, resulting in a remainder of 8.

Continue this process by bringing down additional zeros and repeating the process until the repeating cycle is observed. Long division is instrumental in understanding how fractions convert into repeating decimals.

Division Remainders

Remainders play a key role in the conversion of fractions to decimals, especially when there are repetitive cycles. Each time we divide, the remainder tells us what part of the next digit to consider.

• When dividing 470 by 66, the remainder is 8, indicating how much is left over.
• By adding a zero and making it 80, we see that 66 fits once, leaving a new remainder of 14 after subtraction.
• Continuing this process, another zero makes it 140, and 66 fits in twice, now leaving a remainder of 8.

The repetition of the remainder sequence (8, 80, 140) signals the start of the repeating cycle in the decimal. Understanding when and why remainders repeat is crucial to recognizing repeating decimals, and allows us to denote these numbers accurately using a repeating bar notation, like in \(0.7\overline{12}\).