Problem 26 - Elementary Algebra

Question

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places. \(\frac{5}{8}\)

Step-by-Step Solution

Answer

Answer: The decimal representation is 0.625.

Step 1: Begin division

Divide the numerator (5) by the denominator (8).

Step 2: Add a decimal point

Since we get a remainder, add a decimal point to the numerator (5) and continue the division process.

Step 3: Add a zero after the decimal point

To continue the division process, we need to add a zero after the decimal point.

Step 4: Continue division with the new number

Now, our numerator becomes 50. Divide this new numerator by the denominator (8).

Step 5: Check for termination or rounding

In this case, the division process results in a terminating decimal (0.625). If the decimal were nonterminating, we would round it to 3 decimal places.

Step 6: Write the final answer

Thus, the decimal representation of the fraction \(\frac{5}{8}\) is 0.625.

Key Concepts

Terminating Decimals

When we talk about terminating decimals, we're discussing decimals that come to an end without needing to go on indefinitely. These are decimals that stop at a certain point, meaning no more numbers follow. This usually happens when you divide two numbers and the division has a remainder of zero at some point.
In the fraction to decimal conversion of \( \frac{5}{8} \), the result is 0.625, which is a terminating decimal. Here's how to recognize them:

• If, after some steps in the division process, the division ends exactly without a remainder, you've found a terminating decimal.
• Terminating decimals can also be expressed as being an exact fraction where the denominator has prime factors of only 2 and/or 5.

Recognizing these numbers is important because it guides us on whether we need to round or can stop the process here.

Rounding Decimals

Rounding decimals is a handy way to simplify numbers, especially when dealing with nonterminating decimals. Rounding makes values easier to read and use without significantly affecting the precision necessary for most tasks.
While converting a fraction, if you end up with a long decimal, you may need to round to a specific number of decimal places. Here’s how you can round:

• Identify the place value to which you want to round. If rounding to 3 decimal places, focus on the third digit to the right of the decimal point.
• Look at the digit following it. If this digit is 5 or greater, increase the target digit by one.
• If the digit following is less than 5, keep the target digit unchanged.

Though the fraction \( \frac{5}{8} \) resulted in the terminating decimal 0.625, had it not terminated, you would round according to these rules. This ensures precision and usability in calculations.

Division Process in Fractions

Converting a fraction to a decimal involves a division process. This division is straightforward but needs attention to detail, particularly when fractions don't result in whole numbers.
Here's the process explained for \( \frac{5}{8} \):

• Start dividing the top number (numerator) by the bottom number (denominator). Here, divide 5 by 8.
• Since 5 is less than 8, add a decimal point and a zero, turning it into 50. Continue the division.
• Divide 50 by 8 to get 6, with a remainder of 2. Write 0.6 and continue.
• Add another zero to the remainder, making it 20, and divide again by 8 to get 2, with a remainder of 4. Now you have 0.62.
• Repeat this by adding a zero, turning the remainder into 40, dividing by 8 to get 5, and resulting in the final decimal 0.625.

This step-by-step approach helps in accurately converting a fraction to a decimal, minimizing errors and ensuring clarity in your calculations.