Problem 18 - Algebra 1

Question

Perform any indicated operation. Round the result to the nearest tenth and then to the nearest hundredth. $$ 5.349 \div 46.597 $$

Step-by-Step Solution

Answer

The result of the division is 0.1 when rounded to the nearest tenth and 0.11 when rounded to the nearest hundredth.

Step 1: Division

First perform the division. When 5.349 is divided by 46.597, using calculator if necessary, the result is 0.11476206968996488.

Step 2: Rounding to the nearest tenth

To round the result to the nearest tenth, look at the first digit after the decimal point. If the digit in the tenths place is 5 or greater, round up. If it is less than 5, do not round up. In our case, the first decimal is 1, so rounded to the nearest tenth, our number remains 0.1.

Step 3: Rounding to the nearest hundredth

To round the original result to the nearest hundredth, focus on the second digit after the decimal point. It is 1 which is less than 5, so, when rounding to the nearest hundredth, our number remains 0.11.

Key Concepts

Rounding

Rounding is a mathematical technique used to reduce the number of digits in a number. It helps to simplify numbers for easier interpretation while maintaining a close approximation to the original value. Rounding is commonly performed at various levels of precision, such as rounding to the nearest whole number, tenth, or hundredth.

• To round correctly, identify the digit position to which you're rounding.
• Examine the digit immediately following this position.
• If it's 5 or more, round up the target digit by 1. If it's less than 5, leave the target digit as is.

Rounding is especially useful in scenarios where an exact number is not necessary or to comply with significant figure rules in scientific calculations.

Decimal

Decimals are numbers that contain a decimal point, which separates the whole number part from the fractional part. They are a way to represent fractions using powers of ten, making them an essential part of both basic arithmetic and more advanced mathematics.
Decimals provide precision and flexibility in expressing fractions, which is why they are widely used in various fields like engineering, finance, and everyday transactions.

• The number directly after the decimal is the tenths place, followed by hundredths, thousandths, and so on.
• Understanding the placement of each digit in a decimal is crucial for tasks such as rounding and performing operations like addition, subtraction, multiplication, and division.

Nearest Tenth

Rounding to the nearest tenth involves focusing on the digit in the tenths place of a decimal number. This means considering the first digit after the decimal point. Knowing when to round up or not is key in this process.
For instance, in the number 0.114762, the tenths place is occupied by the digit 1. Since the digit immediately after, which is in the hundredths place, is 1, it dictates that we do not round up. Thus, the number 0.114762 rounded to the nearest tenth becomes 0.1.

• Always observe the digit following the tenths place to decide if rounding up is necessary.
• If rounding is needed, increase the tenths digit by 1.
• Otherwise, keep the tenths digit the same, effectively removing all digits following it.

Nearest Hundredth

Rounding to the nearest hundredth extends the focus to the second digit after the decimal point. In other words, it's about identifying what happens in the hundredths place and using the digit to its right (the thousandths place) to decide rounding needs.
Consider our previous decimal result of 0.114762. Here, the digit in the hundredths place is 1. The digit that follows it is 4, which is less than 5. This indicates that rounding up isn't required, so 0.114762 rounded to the nearest hundredth simplifies to 0.11.

• Check the digit next in line after the hundredths place (which is the thousandths digit).
• If it is 5 or more, raise the hundredths digit by 1.
• Otherwise, the hundredths digit remains unchanged, with all subsequent digits dropped.

Choosing the right rounding method, and understanding decimal positions ensures precision and simplicity where needed.