Problem 30 - Technical Mathematics with Calculus

Question

Round each number to two decimal places. $$2.9573$$

Step-by-Step Solution

Answer

The number rounded to two decimal places is 2.96.

Step 1: Identify the Hundredths Place

Locate the digit in the hundredths place. For the number 2.9573, the hundredths place is occupied by the digit 5.

Step 2: Look at the Digit to the Right

Look at the digit immediately to the right of the hundredths place to determine whether to round up or keep the same. The digit to the right is 7, which is 5 or above.

Step 3: Round the Number

Since the digit to the right is 5 or above, round up the digit in the hundredths place by 1. Thus, 5 becomes 6, making the number 2.96.

Key Concepts

Decimal Places

When it comes to understanding numbers, one fundamental concept is decimal places. These are the positions that appear after the decimal point in a number, and they play a critical role in precision and accuracy in various mathematical settings, ranging from scientific calculations to financial transactions.

For instance, the number 2.9573 has four decimal places, each representing a fraction of a whole: the '9' is in the tenths place (representing nine-tenths), the '5' is in the hundredths place (representing five hundredths), the '7' is in the thousandths place, and the '3' is in the ten-thousandths place. Understanding the position of each digit after the decimal point is key to rounding decimals accurately and effectively.

To practice this concept, try identifying the decimal places in the following numbers: 3.14, 0.001, and 12.345. Recognizing which digit corresponds to which decimal place will strengthen your ability to manipulate and round decimals correctly.

Hundredths Place

Focusing in on the hundredths place, it's the second position after the decimal point and plays a significant role in rounding numbers to two decimal places. In the case of the number 2.9573, the digit ‘5’ is in the hundredths place.

This is especially crucial for financial matters where prices and costs are often calculated to the nearest cent (one-hundredth of a currency unit), making the understanding of the hundredths place incredibly practical. For example, if you were calculating the cost of an item priced at \(3.347, you would need to round the cost to the hundredths place to know your total to the nearest cent, which would be \)3.35.

Practice with these numbers, identifying the digit in the hundredths place: 12.345, 7.8901, and 0.0678. It’s a simple step but is essential for precision in rounding decimals.

Rounding Numbers

Rounding numbers is closely tied to the concept of decimal places and is an operation that helps to simplify numbers, making them easier to work with. The process involves altering a number to the nearest digit at a certain decimal place, based on the digit immediately to its right.

In the step-by-step rounding of 2.9573 to two decimal places, look at the third decimal place (thousandths place), which is the digit '7' in this number, to guide whether to round up or maintain the digit in the hundredths place. Since '7' is 5 or above, the general rule is to round up the digit in the hundredths place, in this case from 5 to 6. Consequently, 2.9573 is rounded to 2.96.

Keep in mind that if the digit to the right is less than 5, you would not round up but rather leave the digit in the hundredths place as it is when rounding to two decimal places. The ability to round numbers effectively is a skill that enhances estimation, simplifies complex calculations, and aids in making more manageable figures for reporting or communicating.