Problem 28 - Elementary and Intermediate Algebra

Question

Write decimal notation for each number. $$ \frac{11}{6} $$

Step-by-Step Solution

Answer

1.8333\bar{3}

Step 1: Understand the Problem

Given the fraction \( \frac{11}{6} \), the goal is to convert this fraction into decimal notation.

Step 2: Perform Division

Divide the numerator (11) by the denominator (6) using long division. When 11 is divided by 6, the quotient is 1 and the remainder is 5. Write down 1 and bring down a zero to continue the division process with the remainder.

Step 3: Continue Division with Remainder

Now divide 50 (which is 5 with a brought down 0) by 6. The quotient is 8 and the remainder is 2. Write down 8 after the decimal point, and bring down another zero.

Step 4: Repeat the Process

Divide 20 by 6. The quotient is 3 and the remainder is 2. This steps reveals repeating decimals. Therefore, continue this process which results in continually repeating decimal digits.

Step 5: Write the Final Decimal Notation

Notice that after step 4, the division repeats the decimal 3 indefinitely. Hence, \( \frac{11}{6} \) in decimal notation is \(1.8333\bar{3}\).

Key Concepts

Fraction to Decimal

Converting a fraction to a decimal involves division. When you see a fraction like \( \frac{11}{6} \), you are essentially being asked to perform the division of 11 by 6.

Here's the simple breakdown:

• The numerator (top number) is 11.
• The denominator (bottom number) is 6.

By dividing 11 by 6, we begin to change the fraction into a decimal. This method is called long division and is very useful for fractions that don't simplify easily.

Long Division

Using long division to convert fractions to decimals helps you handle both simple and complicated fractions.

To understand long division:

• First, write down the division problem: 11 divided by 6.
• 11 divided by 6 gives the quotient 1 with a remainder of 5.

What we have so far is 1. Now, bring down the next digit (0) to join the 5, making it 50. Keep dividing:

• 50 divided by 6 gives the quotient 8. Write 8 next to 1 to start forming 1.8. The remainder is 2.
• Bring down another 0 to make it 20.
• 20 divided by 6 gives the quotient 3. Write 3 after the 8 to form 1.83. The remainder remains 2.

This process repeats itself. Hence, you need to bring down zeroes to keep dividing the remainders, observing if patterns emerge in the quotients.

Repeating Decimals

A repeating decimal happens when you keep getting the same remainder in your long division process. In our example, the division resulted in the decimal \(1.8333\bar{3}\):

• You continuously divide 20 by 6, getting 3 over and over.
• This shows that the digit 3 repeats indefinitely.

In mathematics, repeating decimals are denoted by a bar over the repeating digit(s). So for \( \frac{11}{6} \), the final decimal notation is \(1.8333\bar{3}\). This tells you that 3 keeps repeating on and on.