Problem 9 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$4 \frac{2}{11}$$

Step-by-Step Solution

Answer

The decimal form of \(4 \frac{2}{11}\) is \(4.\overline{18}\).

Step 1: Convert Mixed Number to Improper Fraction

To convert the mixed number \(4 \frac{2}{11}\) to an improper fraction, you multiply the whole number (4) by the denominator (11), and then add the numerator (2). This results in: \(4 \times 11 + 2 = 44 + 2 = 46\). So, \(4 \frac{2}{11}\) becomes \(\frac{46}{11}\).

Step 2: Divide Numerator by Denominator

Next, divide the numerator by the denominator to convert the fraction into a decimal: \(\frac{46}{11}\). Perform the division: \(46 \div 11 = 4.181818...\).

Step 3: Identify Repeating Decimal

Note that the decimal obtained from the division, 4.181818..., shows a repeating pattern. The digits '18' repeat indefinitely.

Step 4: Write the Result with a Repeating Bar

To express the repeating decimal notation, write the decimal as \(4.\overline{18}\). The bar over '18' indicates that these digits repeat.

Key Concepts

Mixed Numbers

Mixed numbers combine whole numbers and fractions. They are written with a whole number part and a fraction, like this: \(4 \frac{2}{11}\).
To convert a mixed number into a decimal, we first turn it into an improper fraction.

• This involves multiplying the whole number by the denominator of the fraction.
• Then, add the numerator of the fraction to the result.

For instance, in the mixed number \(4 \frac{2}{11}\):
Multiply 4 by 11 to get 44.
Now, add 2, which is the numerator, to get 46.
Thus, the mixed number \(4 \frac{2}{11}\) is the same as the improper fraction \(\frac{46}{11}\).

Understanding mixed numbers and being able to convert them into improper fractions is crucial when you want to change them into decimals. It sets the foundation for the next steps in conversion.

Repeating Decimals

Repeating decimals occur when a division of numbers results in a decimal that has one or more digits repeating infinitely.
For example, when we divide \(46\) by \(11\), the result is the decimal \(4.181818...\), where '18' repeats endlessly.

• Such decimals are often expressed using a line or bar over the repeating digits.
• This bar efficiently indicates that a sequence of digits recurs.

In our example, we write the repeating decimal as \(4.\overline{18}\), placing the bar over '18' to show it continues.

Repeating decimals can often appear when converting fractions to decimals, especially when the denominator is not a factor of 10. Knowing how to properly write repeating decimals can help in clarity and prevent confusion in mathematical notation.

Improper Fractions

An improper fraction is one where the numerator is larger than or equal to the denominator.
This means the fraction represents a value greater than or equal to one.
For example, the fraction \(\frac{46}{11}\) is improper because 46 is greater than 11.

• Improper fractions are useful when converting mixed numbers to decimals.
• They simplify the calculation as they eliminate whole parts of fractions.

First, take the improper fraction \(\frac{46}{11}\).
By dividing 46 by 11 to get 4.181818..., we observe it reflects its original mixed number form.

Understanding improper fractions and their conversions provides a better grasp of fractions and decimals. They play an essential role in mathematics by linking various numerical forms and assisting in performing computations efficiently.