Calculating 2.35 Multiplied By 2/3 A Step-by-Step Guide

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In this comprehensive guide, we will meticulously walk through the calculation of 2.35 multiplied by 2/3. This seemingly simple mathematical problem touches upon fundamental concepts of decimal and fraction arithmetic, making it a valuable exercise for students and anyone looking to brush up on their math skills. We will break down each step, providing clear explanations and insights to ensure a thorough understanding of the process. By the end of this article, you'll not only know the answer but also grasp the underlying principles that make this calculation work.

Understanding the Problem: 2.35 * (2/3)

The core of our task lies in accurately multiplying a decimal number, 2.35, by a fraction, 2/3. This requires us to first understand how decimals and fractions interact during multiplication. Decimals represent numbers that are not whole, using a base-10 system, while fractions express parts of a whole using a numerator and a denominator. To effectively perform this multiplication, we need to ensure both numbers are in a compatible format. There are two primary approaches we can take: either convert the decimal to a fraction or perform the multiplication directly using decimal arithmetic principles. Both methods will lead us to the same correct answer, but understanding each approach can enhance our problem-solving versatility. The process involves several key steps, including converting the decimal to a fraction (if needed), multiplying the fractions or decimals, and simplifying the result. This might also involve converting the final answer back to a decimal if required, allowing for a comprehensive understanding of the relationship between these two forms of numerical representation.

Converting the Decimal to a Fraction

The first step in solving this problem is to convert the decimal 2.35 into a fraction. This conversion is crucial because it allows us to perform the multiplication using the rules of fraction arithmetic, which are straightforward and well-defined. To convert 2.35 into a fraction, we first recognize that the decimal has two digits after the decimal point, indicating that it represents hundredths. Therefore, we can express 2.35 as 235/100. This fraction represents the same value as the decimal but is now in a form that we can easily work with when multiplying by another fraction. Once we have the fraction 235/100, we should consider simplifying it to its lowest terms. Simplification makes the subsequent multiplication and simplification steps easier to manage. To simplify 235/100, we need to find the greatest common divisor (GCD) of 235 and 100. Both numbers are divisible by 5, so we can divide both the numerator and the denominator by 5. This gives us 47/20, which is the simplified fractional form of 2.35. Understanding this conversion process is vital as it forms the foundation for solving the multiplication problem.

Multiplying the Fractions: 47/20 * 2/3

Now that we've converted 2.35 to its fractional equivalent, 47/20, we can proceed with the multiplication by 2/3. Multiplying fractions is a fundamental operation in arithmetic, and it involves a straightforward process: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. In this case, we multiply 47 (the numerator of the first fraction) by 2 (the numerator of the second fraction), resulting in 94. Then, we multiply 20 (the denominator of the first fraction) by 3 (the denominator of the second fraction), which gives us 60. Thus, the product of the two fractions is 94/60. This fraction represents the initial result of the multiplication, but it is not yet in its simplest form. The next crucial step is to simplify this fraction to its lowest terms. Simplifying fractions is essential because it presents the result in the most concise and understandable form. It also makes it easier to compare the result with other fractions or to convert it back to a decimal if necessary. Therefore, we need to find the greatest common divisor (GCD) of 94 and 60 to simplify the fraction further.

Simplifying the Resulting Fraction

After multiplying the fractions, we arrived at the fraction 94/60. To simplify this fraction, we need to find the greatest common divisor (GCD) of 94 and 60. The GCD is the largest number that divides both 94 and 60 without leaving a remainder. One way to find the GCD is to list the factors of each number and identify the largest factor they have in common. The factors of 94 are 1, 2, 47, and 94, while the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. By comparing these lists, we can see that the greatest common factor is 2. Once we have identified the GCD, we can divide both the numerator and the denominator of the fraction by this value. Dividing 94 by 2 gives us 47, and dividing 60 by 2 gives us 30. Therefore, the simplified fraction is 47/30. This simplified fraction is now in its lowest terms, meaning that the numerator and denominator have no common factors other than 1. This is an important step in presenting the answer in the most understandable form. The fraction 47/30 represents the exact result of multiplying 2.35 by 2/3, expressed as a simplified fraction.

Converting the Improper Fraction to a Mixed Number

Our simplified fraction, 47/30, is an improper fraction because the numerator (47) is greater than the denominator (30). While improper fractions are perfectly valid mathematical expressions, it is often more intuitive to express them as mixed numbers. A mixed number consists of a whole number part and a proper fraction part, making it easier to visualize the quantity the fraction represents. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part. The denominator remains the same. In the case of 47/30, we divide 47 by 30. The quotient is 1, which means the whole number part of our mixed number is 1. The remainder is 17, which becomes the numerator of the fractional part. The denominator remains 30. Therefore, the mixed number equivalent of 47/30 is 1 17/30. This mixed number representation provides a clear sense of the value: it is slightly more than 1. Converting to a mixed number is not always necessary, but it can be helpful for understanding the magnitude of the result, especially in practical applications where a mixed number can provide a more intuitive sense of quantity.

Converting the Fraction to a Decimal (Optional)

While we have successfully calculated the result as a simplified fraction (47/30) and a mixed number (1 17/30), there are situations where expressing the answer as a decimal is more practical or required. Converting a fraction to a decimal involves dividing the numerator by the denominator. In our case, we need to divide 47 by 30. This division will yield a decimal number, which may be a terminating decimal or a repeating decimal. Performing the division, 47 ÷ 30, we get 1.5666... The digit 6 repeats indefinitely, indicating that this is a repeating decimal. When dealing with repeating decimals, it is common practice to round the decimal to a certain number of decimal places, depending on the desired level of precision. For instance, rounding to two decimal places gives us 1.57. Rounding to three decimal places gives us 1.567. The level of precision required depends on the context of the problem and the level of accuracy needed. Converting to a decimal provides an alternative representation of the result, which can be useful for comparison with other decimal values or for practical applications where decimal measurements are preferred.

Possible Answers and The Correct Solution

Now, let's examine the possible answers provided in the original problem: 7/30, 7/15, and 27/30. We have already calculated the correct answer as 47/30. Comparing this to the given options, it is clear that none of them match our calculated result. This discrepancy highlights the importance of carefully working through each step of the calculation to ensure accuracy. It also underscores the value of understanding the underlying mathematical principles, rather than simply guessing or relying on intuition. The correct solution, 47/30 (or 1 17/30 as a mixed number, or approximately 1.57 as a decimal), is significantly different from the options provided, emphasizing the need for a thorough and methodical approach to solving mathematical problems. This process not only yields the correct answer but also reinforces understanding and builds confidence in mathematical abilities. In the next section, we will summarize the steps we took to arrive at the solution and highlight the key concepts involved.

Conclusion: Key Takeaways and Problem-Solving Strategies

In conclusion, we have successfully calculated the result of multiplying 2.35 by 2/3, arriving at the answer 47/30, which can also be expressed as the mixed number 1 17/30 or the decimal approximation 1.57. This calculation involved several key steps, each of which highlights important mathematical principles. First, we converted the decimal 2.35 to its fractional equivalent, 235/100, and simplified it to 47/20. This step demonstrated the relationship between decimals and fractions and the importance of simplification. Next, we multiplied the fractions 47/20 and 2/3, resulting in 94/60. We then simplified this fraction to its lowest terms, 47/30, by finding the greatest common divisor of the numerator and denominator. This process reinforced the rules of fraction multiplication and simplification. We also converted the improper fraction 47/30 to the mixed number 1 17/30, illustrating how to represent fractions in different forms. Finally, we converted the fraction to a decimal approximation, 1.57, demonstrating the connection between fractions and decimals and the concept of rounding. This comprehensive approach not only provided the correct answer but also reinforced fundamental mathematical concepts and problem-solving strategies. By understanding each step and the underlying principles, we can confidently tackle similar problems in the future. This exercise serves as a valuable reminder of the importance of accuracy, attention to detail, and a thorough understanding of mathematical concepts in problem-solving.