Calculating 4 Divided By 5/7 Expressed As A Fraction

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Understanding the Division of Fractions

In the realm of mathematics, dividing by a fraction might initially seem perplexing, but it's a fundamental operation with a straightforward principle. The core concept lies in understanding that dividing by a fraction is equivalent to multiplying by its reciprocal. This might sound complex, but let's break it down. The reciprocal of a fraction is simply flipping the numerator and the denominator. For instance, the reciprocal of 57\frac{5}{7} is 75\frac{7}{5}. This simple maneuver transforms division into multiplication, an operation we're often more comfortable with. When we encounter a problem like 4Γ·574 \div \frac{5}{7}, the first step is to recognize that we're dividing the whole number 4 by the fraction 57\frac{5}{7}. To perform this division, we convert it into multiplication by the reciprocal, turning our problem into 4Γ—754 \times \frac{7}{5}. Now, we're dealing with the multiplication of a whole number by a fraction, which is a more manageable task. This conversion is not just a trick; it's rooted in the mathematical definition of division as the inverse operation of multiplication. Understanding this principle is crucial for handling various fraction-related problems, from simple arithmetic to more complex algebraic equations. The reciprocal transformation is a powerful tool in simplifying and solving divisions involving fractions, ensuring accuracy and efficiency in mathematical calculations. Mastering this concept provides a solid foundation for tackling more advanced topics in mathematics.

Step-by-Step Calculation

To effectively calculate 4Γ·574 \div \frac{5}{7}, we embark on a step-by-step journey, making sure each stage is clearly understood. The initial challenge is to transform the division problem into a multiplication problem using the reciprocal. As we've established, dividing by a fraction is the same as multiplying by its reciprocal. So, 4Γ·574 \div \frac{5}{7} becomes 4Γ—754 \times \frac{7}{5}. This transformation is the cornerstone of solving such problems. Next, we need to express the whole number 4 as a fraction. Any whole number can be written as a fraction by placing it over a denominator of 1. Thus, 4 is equivalent to 41\frac{4}{1}. Our expression now looks like 41Γ—75\frac{4}{1} \times \frac{7}{5}. The multiplication of fractions is a straightforward process: we multiply the numerators together and the denominators together. In this case, we multiply 4 (the numerator of the first fraction) by 7 (the numerator of the second fraction), resulting in 28. Similarly, we multiply 1 (the denominator of the first fraction) by 5 (the denominator of the second fraction), giving us 5. So, the result of the multiplication is the fraction 285\frac{28}{5}. This fraction represents the solution to our original division problem. It's a clear and concise answer, but our journey isn't quite over yet. We need to ensure that our answer is in its simplest form, adhering to the common practice in mathematics of expressing fractions in their lowest terms. This involves checking if the fraction can be further simplified.

Expressing the Answer in Lowest Terms

After performing the calculation 4Γ·574 \div \frac{5}{7} and arriving at the fraction 285\frac{28}{5}, the final step is to express this answer in its lowest terms. This is a crucial aspect of simplifying fractions and presenting them in their most concise form. To determine if 285\frac{28}{5} is in its lowest terms, we need to identify the greatest common divisor (GCD) of the numerator (28) and the denominator (5). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. If the GCD is 1, the fraction is already in its lowest terms. In this case, the factors of 28 are 1, 2, 4, 7, 14, and 28, while the factors of 5 are 1 and 5. The only common factor between 28 and 5 is 1, which means their GCD is 1. Therefore, the fraction 285\frac{28}{5} is indeed in its lowest terms. Since the fraction cannot be simplified further, we can confidently present it as the final answer. However, it's worth noting that 285\frac{28}{5} is an improper fraction because the numerator is greater than the denominator. While 285\frac{28}{5} is a perfectly acceptable answer, it can also be expressed as a mixed number, which combines a whole number and a proper fraction. Converting 285\frac{28}{5} to a mixed number involves dividing 28 by 5. The quotient is 5, and the remainder is 3. This means that 285\frac{28}{5} is equivalent to 5 whole units and 35\frac{3}{5}. Thus, the mixed number representation of 285\frac{28}{5} is 5355\frac{3}{5}. Both 285\frac{28}{5} and 5355\frac{3}{5} are valid answers, but the context of the problem or specific instructions might indicate a preference for one form over the other. In conclusion, the fraction 285\frac{28}{5} is already in its lowest terms, and it accurately represents the result of the division. The process of expressing fractions in their lowest terms ensures clarity and precision in mathematical communication.

Therefore, 4Γ·57=2854 \div \frac{5}{7} = \frac{28}{5}.