Problem 21 - Calculate With Confidence

Question

Solve the following. Carry division to the hundredths place where necessary. $$472.4 \times 0.002=$$

Step-by-Step Solution

Answer

0.94

Step 1: Understand the Problem

We are asked to multiply the number 472.4 by 0.002 and ensure our answer is accurate to the hundredths decimal place. This means our final answer should have two digits after the decimal point.

Step 2: Set Up the Multiplication

The problem is to compute the multiplication of two numbers: 472.4 and 0.002. We'll write it out as follows:\[ 472.4 \times 0.002 \]

Step 3: Perform the Multiplication

To multiply these numbers, first ignore the decimal points and multiply them as whole numbers.Convert 472.4 to 47240 (shifting the decimal three places) and 0.002 to 2 (shifting the decimal three places in the opposite direction), then multiply:\[ 47240 \times 2 = 94480 \]

Step 4: Adjust the Decimal Point

Since we shifted the decimal points in Step 3, we need to account for this change. There were a total of five decimal places shifted (three from 472.4 and three from 0.002, combined six, but one cancels out when dealing with multiplication of decimals).Returning these five decimal places to the result 94480 gives:\[ 0.94480 = 0.9448 \]

Step 5: Round to the Hundredths Place

Now, take the calculated number 0.9448 and round it to the nearest hundredth place. The digit in the hundred place is 4, and the next digit is 8, which means we round up.Thus, the rounded answer is:\[ 0.94 \]

Step 6: Conclusion

The product of 472.4 and 0.002, rounded to the hundredths place, is 0.94.

Key Concepts

Rounding Decimals

Rounding decimals is an essential skill that helps us work with more manageable numbers, especially in real-life situations. It involves adjusting a number to a specified degree of precision. To round a decimal, follow these steps:

• Identify the place value to which you need to round. Common values are tenths, hundredths, etc.
• Look at the digit immediately to the right of the place value you're rounding to. This digit determines whether you 'round up' or 'round down'.
• If this digit is 5 or greater, increase the digit at the place value by one (round up). If it’s less than 5, keep the digit the same (round down).
• Remove all the digits to the right of your new rounded digit.

Rounding makes numbers simpler and more practical for quick calculations or reporting. It’s important to consider the context when rounding is appropriate or necessary.

Decimal Places

Understanding decimal places is crucial when working with decimals. Decimal places refer to the number of digits to the right of the decimal point in a number. They show the precision of the number.

• The first digit to the right of the decimal is called the 'tenths' place.
• The second digit is the 'hundredths' place.
• The third digit is the 'thousandths' place, and so on.

Decimal places are important in various fields such as finance, engineering, and science where precision is critical. They help in representing exact values rather than rounded estimations. It is also key when aligning numbers to ensure proper addition, subtraction, multiplication, or division.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics, consisting of addition, subtraction, multiplication, and division. In terms of multiplication involving decimals:

• Ignore the decimal points initially and multiply the numbers as if they are whole numbers.
• Count the total number of decimal places in both the numbers you're multiplying.
• After multiplying the whole numbers, place the decimal in the result. The number of decimal places in the result should match the total number observed in the initial numbers.

This method maintains the precision needed in calculations and helps in producing accurate results. Understanding this process is vital for dealing with any calculations involving decimals.

Mathematics Education

Mathematics education focuses on equipping students with the skills they need to solve problems and think critically. Mastering concepts such as decimal multiplication is integral to mathematics learning.

• It enhances arithmetic skills, crucial for academic success and everyday life.
• Learning precise calculation methods builds confidence and reduces errors.
• Real-world applications, like finances and measurements, often require a solid understanding of decimal manipulation.

Teachers play a significant role by providing practical examples and exercises, helping students to apply these concepts in real-life scenarios. Good mathematics education fosters a deeper understanding and prepares students for advanced topics in sciences and beyond.