Problem 55 - Prealgebra

Question

For Exercises 55 and \(56,\) use the following information. Write \(\frac{5}{6}\) as a decimal rounded to the nearest hundredth.

Step-by-Step Solution

Answer

\( \frac{5}{6} \) as a decimal rounded to the nearest hundredth is 0.83.

Step 1: Understanding the Problem

We need to convert the fraction \( \frac{5}{6} \) into a decimal number and round it to the nearest hundredth. The hundredth place is the second digit to the right of the decimal point.

Step 2: Convert Fraction to Decimal

Divide 5 by 6 to convert \( \frac{5}{6} \) into a decimal. Performing the division 5 ÷ 6 gives us 0.8333... which is a repeating decimal.

Step 3: Identify the Decimal Places

Locate the first two decimal places of 0.8333..., which are 8 (tenths) and 3 (hundredths). We will consider the third decimal place (which is also 3) for rounding.

Step 4: Round Off to Nearest Hundredth

To round 0.8333... to the nearest hundredth, look at the third decimal place, which is 3. Since it is less than 5, we do not increase the hundredth place. Hence, 0.83 is the rounded number.

Key Concepts

Rounding Decimals

Rounding decimals means adjusting the number to its most convenient form, often to make it simpler or to meet a certain level of precision. When dealing with decimals, the most common places to round are the tenths, hundredths, or thousandths. For example, when you see a number such as 0.8333, it includes many repeating numbers. To round it:

• Check the digit just to the right of your desired place value. For instance, if rounding to hundredths, check the third digit after the decimal point.
• If this digit is 5 or greater, round the last digit you keep up by one. If it's less than 5, the digit remains the same.
• Thus, rounding 0.8333... to the nearest hundredth results in 0.83, because the third decimal place is a 3, which is less than 5.

Repeating Decimals

Repeating decimals occur when a number does not resolve into a finite decimal representation. Instead, one or more digits repeat infinitely. When dividing 5 by 6, you get 0.8333... with the digit '3' repeating. To identify a repeating decimal:

• Perform the division operation.
• Notice if the same digit or group of digits repeats endlessly.

The decimal for the fraction \( \frac{5}{6} \) is written with a bar over the repeating digit: 0.8\overline{3}. Understanding repeating decimals helps in precise calculation and representation in cases where rounding isn't suitable.

Basic Arithmetic Operations

Converting a fraction to a decimal is an example of division, one of the basic arithmetic operations. To divide 5 by 6 and convert \(\frac{5}{6}\) into a decimal:

• Set up the division problem where 5 is the dividend and 6 is the divisor.
• Perform the division by determining how many times 6 can be subtracted from 5. Here, you'll notice 6 doesn't fully divide into 5, so add a decimal.
• Add zero after zero to the dividend as needed, and continue the division.
• The result is 0.8333..., illustrating that basic division can yield a repeating decimal.

Understanding these basic arithmetic operations is essential for accurately converting fractions to decimals and tackling more complex math problems.