Calculating The Product Of 0.025 And 1.5 A Step By Step Guide
Finding the product of decimal numbers might seem daunting at first, but with a clear understanding of the underlying principles and a step-by-step approach, it becomes a manageable task. In this comprehensive guide, we will delve into the process of multiplying 0.025 and 1.5, exploring the concepts involved, and arriving at the correct solution. We will break down each step, providing detailed explanations and examples to ensure clarity. Furthermore, we will discuss the significance of scientific notation in expressing very small or very large numbers, and how it relates to the given options. This guide aims to provide a thorough understanding of the multiplication process, empowering you to tackle similar problems with confidence and accuracy.
Breaking Down the Problem: Multiplying Decimals
When we multiply decimals, we are essentially combining the concepts of decimal numbers and multiplication. Decimals represent fractional parts of whole numbers, and multiplication is the process of repeated addition. To multiply 0.025 and 1.5, we need to consider the place values of each digit and how they interact during multiplication. The number 0.025 can be interpreted as 25 thousandths, while 1.5 represents one and five tenths. Multiplying these numbers involves distributing each digit of one number across the digits of the other, considering their respective place values. This process can be simplified by initially ignoring the decimal points and treating the numbers as whole numbers, then adjusting the decimal point in the final result. Understanding this fundamental principle is crucial for accurately multiplying decimals and avoiding common errors. In the subsequent sections, we will demonstrate this process step-by-step, highlighting the key considerations and techniques involved. This approach will not only help you solve this particular problem but also equip you with a broader understanding of decimal multiplication.
Step-by-Step Multiplication Process
The process of multiplying 0.025 and 1.5 involves several key steps. First, we ignore the decimal points and treat the numbers as whole numbers, which gives us 25 and 15. Then, we perform the multiplication as we would with whole numbers:
25
* 15
----
125
+25
----
375
Next, we need to account for the decimal points. In the original numbers, 0.025 has three decimal places (three digits after the decimal point), and 1.5 has one decimal place. So, in total, there are four decimal places (3 + 1 = 4). This means we need to place the decimal point in our result (375) four places from the right. Counting four places from the right in 375 gives us 0.0375. Therefore, the product of 0.025 and 1.5 is 0.0375. This meticulous step-by-step approach ensures accuracy and avoids errors that can occur when dealing with decimal numbers. It's important to double-check the number of decimal places to ensure the final answer is correct. In the next section, we'll explore how this result aligns with the provided options and discuss the concept of scientific notation.
Understanding Scientific Notation
Scientific notation is a way of expressing numbers, especially very large or very small numbers, in a concise and standardized format. It is written as a × 10^b, where a is a number between 1 and 10 (but not including 10), and b is an integer (positive, negative, or zero). This notation is particularly useful in scientific and engineering contexts where dealing with extremely large or small values is common. For instance, the speed of light in a vacuum is approximately 299,792,458 meters per second, which can be written in scientific notation as 2.99792458 × 10^8 m/s. Similarly, the size of an atom is incredibly small, and scientific notation allows us to express these dimensions in a more manageable way. The exponent b indicates the number of places the decimal point needs to be moved to obtain the standard decimal representation. A positive exponent means the decimal point is moved to the right, while a negative exponent means it is moved to the left. In the context of our problem, understanding scientific notation helps us interpret the answer choices and express our result in a different form if needed. In the next section, we will convert our calculated product into scientific notation to compare it with the given options.
Converting 0.0375 to Scientific Notation
To convert 0.0375 to scientific notation, we need to express it in the form a × 10^b, where 1 ≤ |a| < 10 and b is an integer. In this case, we need to move the decimal point two places to the right to get the number 3.75, which falls within the required range. Moving the decimal point two places to the right corresponds to dividing by 100, or 10^2. To compensate for this, we multiply by 10^-2. Therefore, 0.0375 can be written as 3.75 × 10^-2 in scientific notation. This conversion is crucial for comparing our calculated result with the given options, which are often presented in scientific notation. Understanding how to convert between decimal notation and scientific notation is a valuable skill in mathematics and science, allowing for easier manipulation and comparison of numbers. In the following section, we will analyze the answer choices and identify the one that matches our result in scientific notation.
Analyzing the Answer Choices
Now that we have calculated the product of 0.025 and 1.5 and converted it to scientific notation, we can analyze the provided answer choices to determine the correct one. The choices are:
A. 3.75 × 10^-2 B. 3.75 × 10^2 C. 0.375
Our calculated result, 0.0375, is equivalent to 3.75 × 10^-2 in scientific notation. Comparing this with the answer choices, we can see that option A, 3.75 × 10^-2, matches our result. Option B, 3.75 × 10^2, is incorrect because it represents 375, which is significantly larger than our calculated product. Option C, 0.375, is also incorrect because it is ten times larger than our calculated product of 0.0375. Therefore, the correct answer is A. This process of elimination and comparison highlights the importance of accurate calculation and conversion to scientific notation when dealing with multiple-choice questions. In the concluding section, we will summarize our findings and reiterate the key concepts learned.
Conclusion: The Product and Its Significance
In conclusion, we have successfully found the product of 0.025 and 1.5, which is 0.0375. We also converted this result to scientific notation, expressing it as 3.75 × 10^-2. Through a step-by-step approach, we demonstrated the process of multiplying decimals, emphasizing the importance of accurately placing the decimal point in the final result. We also explored the concept of scientific notation, highlighting its utility in representing very small numbers in a concise format. By analyzing the answer choices, we confirmed that option A, 3.75 × 10^-2, is the correct answer. This exercise not only reinforces our understanding of decimal multiplication but also enhances our ability to work with scientific notation. The ability to accurately multiply decimals and convert them to scientific notation is a fundamental skill in mathematics and has practical applications in various fields, including science, engineering, and finance. Mastering these concepts will undoubtedly contribute to your problem-solving abilities and overall mathematical proficiency.
