Problem 32 - Prealgebra

Question

Convert the given fraction to a repeating decimal. Use the "repeating bar” notation. \(\frac{37}{18}\)

Step-by-Step Solution

Answer

The decimal representation is \(2.05\overline{5}\).

Step 1: Perform Long Division

To convert the fraction \( \frac{37}{18} \) into a decimal, we start by performing long division. Divide 37 by 18. 18 goes into 37 two times (\(2 \times 18 = 36\)), leaving a remainder of 1.

Step 2: Continue Division with a Decimal

Bring down a zero to make 10 since there's a remainder. Now divide 10 by 18. 18 goes into 10 zero times, so place 0 after the decimal point and consider 100 next.

Step 3: Identify the Repeating Section

Continue division by bringing down another zero, making it 100. 18 goes into 100 five times (since \(5 \times 18 = 90\)), leaving a remainder of 10. This indicates the number is repeating.

Step 4: Complete the Repeating Pattern

Notice that when you bring another 0 down to form 100 again, it results in the same calculation as before, indicating the pattern will continue. Hence, the decimal is \(2.0555...\) with 5 repeating.

Step 5: Express the Decimal with Repeating Bar Notation

Finally, write the decimal with repeating part using the repeating bar notation. The fraction \( \frac{37}{18} \) is \(2.05\overline{5}\) where the 5 is the repeating part.

Key Concepts

Understanding Long Division

Long division is a method for dividing large numbers and is also crucial when you want to convert a fraction to a decimal. To perform long division with fractions like \( \frac{37}{18} \), you start by dividing the numerator, which is 37, by the denominator, which is 18. Think of long division as a series of simple steps:

• Divide: Determine how many times 18 can fit into 37. In this case, it fits twice.
• Multiply: Multiply 2 (your result) by 18, which gives you 36.
• Subtract: Subtract 36 from 37, leaving a remainder of 1.
• Bring Down: Since we have a remainder, bring down a zero to continue dividing.

These steps are repeated, especially if you find that you keep having remainders, to see if more decimals will repeat. Long division helps identify whether converting a fraction will lead to a repeating decimal or not.

Fraction Conversion Explained

When converting a fraction like \( \frac{37}{18} \) to a decimal, you often use long division. But understanding why this works can be equally important. Conversion of fractions into decimals involves determining how many times the denominator can be divided into the numerator and then tracking the remainders.Here’s how fraction conversion helps:

• Fraction parts: It naturally shows how much more is left after each division.
• Decimal places: Each time you bring down a zero, you’re essentially moving into the next decimal place (tenths, hundredths, etc.).
• Repeating decimals: If the remainder repeats, you’ve found a repeating decimal! This tells you that a section of the decimal will continue indefinitely.

Fraction to decimal conversion is a visual and numerical way to reconceptualize the fraction's value. Sometimes, it can reveal interesting patterns, like repeating decimals, represented by a bar notation!

Decimal Notation Insights

Decimal notation is the representation of fractions in a base-10 format. When a fraction like \( \frac{37}{18} \) is translated into decimals, the notation helps to eloquently express the idea of repeating sequences.Decimal notation:

• Shows precision: It gives us a way to express non-whole numbers accurately.
• Uses repeating bar: For repeating decimals, a horizontal line (or bar) is placed over the digits that repeat. For our example, this is shown as \(2.05\overline{5}\).
• Illustrates endless repetition: The bar signifies that the 5 repeats indefinitely. This notation is efficient for communicating a number that would otherwise be long and cumbersome.

Understanding decimal notation is key to interpreting repeating decimals and appreciating the detailed depiction they provide of fractional values.