Problem 38 - Prealgebra

Question

Carry out cach of the following divisions only so far as needed to round the results to the nearest hundredth. $$16 \div 6$$

Step-by-Step Solution

Answer

16 divided by 6, rounded to the nearest hundredth, is 2.67.

Step 1: Set Up Division

Write the division problem as a fraction: \(16 \div 6 = \frac{16}{6}\). This can also be expressed as a long division where 16 is the dividend and 6 is the divisor.

Step 2: Perform Long Division

Carry out the division of 16 by 6. You will find that 6 goes into 16 two times (since 6 times 2 equals 12) with a remainder of 4.

Step 3: Continue Division with Decimal Places

Bring down a zero, making it 40. Divide 40 by 6, which gives 6 (since 6 times 6 equals 36), leaving a remainder of 4 again. Bring down another zero, making it 40 again.

Step 4: Determine Decimal Places

Continuing from where we left off. 40 divided by 6 is 6, remainder 4. Since the remainder repeats, you get a repeating decimal of 2.666....

Step 5: Round to the Nearest Hundredth

The decimal 2.666... goes on infinitely, but rounds to 2.67 when rounded to the nearest hundredth, because the digit in the thousandths place is 6, which rounds the hundredths up.

Key Concepts

Understanding Long Division

Long division is a methodical process used to divide large numbers that involves dividing, multiplying, and subtracting steps. When we divide 16 by 6 using long division:

• We start by seeing how many times 6 fits into 16. It fits 2 times, since 6 times 2 equals 12.
• Subtract 12 from 16 to leave a remainder of 4.
• Next, to get a more precise answer, we bring down a zero, turning the remainder into 40, and then divide 40 by 6.
• This time, 6 fits into 40 a total of 6 times, since 6 times 6 equals 36.
• Subtract 36 from 40, leaving a remainder of 4 once more. This process showing the division into decimals highlights the relationship between the divisor, dividend, and remainder.

Long division helps us determine decimal values and can be repeated until we achieve the desired precision.

Exploring Repeating Decimals

When division results in a decimal that endlessly continues without stopping, it leads to what we call a repeating decimal. For example, when dividing 16 by 6 using long division, we observe the pattern 2.666....

• This means after the decimal point, the number 6 keeps repeating infinitely.
• The repeating sequence is usually represented by placing a line over the digits that repeat, like this: \(2.\overline{6}\).
• Repeating decimals arise when the remainder isn't zero and continues cycling through a pattern as you carry on with the division.

Understanding repeating decimals is helpful to anticipate rounding, as it indicates where the decimal values continue indefinitely.

Rounding to the Nearest Hundredth

Rounding decimals is a key skill necessary to simplify otherwise lengthy numbers into manageable figures. The goal of rounding is to find the nearest desired place value.

• The problem asks us to round \(2.666...\) to the nearest hundredth. The hundredths place in \(2.666...\) is the second 6.
• To decide whether to round up or stay the same, look at the digit immediately after the hundredths place—this is referred to as the thousandths place.
• If this digit is 5 or more, you round up the hundredths digit by 1. In this case, the number in the thousandths place is 6, so we round the 6 in the hundredths place up, transforming \(2.666...\) into \(2.67\).

Rounding ensures we have a clearer, more concise representation of numbers for easier calculations and communication.