Solving $5^1 / 4 \div 2^2 / 7$: A Step-by-Step Guide

This article aims to break down the calculation $5^1 / 4 \div 2^2 / 7$ step by step, ensuring a clear understanding of the mathematical principles involved. We will explore the order of operations, the conversion of mixed numbers to improper fractions, and the rules of division when dealing with fractions. By the end of this explanation, you should be able to confidently tackle similar mathematical problems and grasp the underlying concepts.

Breaking Down the Expression

The given expression is $5^1 / 4 \div 2^2 / 7$. At first glance, it may seem confusing due to the combination of whole numbers, fractions, and the division operation. However, by systematically applying the rules of arithmetic, we can simplify it. The key here is to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this particular expression, we have exponents, division, and fractions to contend with. Therefore, it's vital to address each component in a logical sequence.

Step 1: Evaluate the Exponents

The first step in simplifying the expression is to evaluate the exponents. We have $5^1$ and $2^2$.

• $5^1$ simply means 5 raised to the power of 1, which is 5.
• $2^2$ means 2 raised to the power of 2, which is $2 \times 2 = 4$.

So, the expression now becomes $5 / 4 \div 4 / 7$. By addressing the exponents first, we have reduced the complexity of the original expression, making it easier to work with. This is a fundamental step in solving mathematical problems that involve multiple operations. The order in which we perform these operations can significantly affect the outcome, highlighting the importance of adhering to established mathematical rules.

Step 2: Convert Mixed Numbers (If Any) to Improper Fractions

In this particular problem, we don't have traditional mixed numbers like $2 \frac{1}{2}$. Instead, we have whole numbers and fractions. The expression $5 / 4$ and $4 / 7$ are already in fractional form, so this step is straightforward. However, it's crucial to understand how to convert mixed numbers to improper fractions in general, as it is a common step in many similar problems. A mixed number consists of a whole number and a fraction (e.g., $3 \frac{1}{2}$). To convert it to an improper fraction, you multiply the whole number by the denominator of the fraction, add the numerator, and then place the result over the original denominator.

For instance, if we had $2 \frac{1}{3}$, we would calculate $(2 \times 3) + 1 = 7$, and then write it as $\frac{7}{3}$. This conversion is essential because it allows us to perform arithmetic operations, especially multiplication and division, more easily with fractions. Improper fractions are easier to manipulate in calculations compared to mixed numbers. In our case, since we already have $5 / 4$ and $4 / 7$, we can proceed directly to the division step, as these are already in the required fractional form.

Step 3: Dividing Fractions

Now we come to the core of the problem: dividing the fractions. The expression is currently $5 / 4 \div 4 / 7$. Dividing fractions can sometimes seem tricky, but there's a simple rule to follow: invert and multiply. This means we flip the second fraction (the divisor) and change the division operation to multiplication. So, $5 / 4 \div 4 / 7$ becomes $5 / 4 \times 7 / 4$.

Flipping the second fraction, $4 / 7$, gives us its reciprocal, which is $7 / 4$. By changing the division to multiplication, we transform the problem into a more manageable form. This method works because division is the inverse operation of multiplication. When you divide by a fraction, it's the same as multiplying by its reciprocal. This principle is a cornerstone of fraction arithmetic and is crucial for solving various mathematical problems.

Step 4: Multiply the Fractions

After inverting and multiplying, we now have $5 / 4 \times 7 / 4$. To multiply fractions, we simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we calculate:

• Numerator: $5 \times 7 = 35$
• Denominator: $4 \times 4 = 16$

This gives us the fraction $35 / 16$. Multiplying fractions is a straightforward process once you understand the rule. The resulting fraction represents the product of the two original fractions. In this case, $35 / 16$ is the result of multiplying $5 / 4$ by $7 / 4$. This step is crucial in arriving at the final answer, and it highlights the importance of accurately performing multiplication operations.

Step 5: Simplify the Fraction (If Possible)

Our current result is $35 / 16$. Now, we need to check if this fraction can be simplified. Simplification involves reducing the fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. In this case, the numerator is 35, and the denominator is 16. The factors of 35 are 1, 5, 7, and 35. The factors of 16 are 1, 2, 4, 8, and 16. The only common factor between 35 and 16 is 1, which means the fraction is already in its simplest form.

However, we can also express this improper fraction as a mixed number to provide a more intuitive understanding of its value. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same.

• $35 \div 16 = 2$ with a remainder of 3.

So, $35 / 16$ can be written as the mixed number $2 \frac{3}{16}$. This mixed number representation gives us a better sense of the magnitude of the fraction. We can see that it is slightly more than 2 whole units.

Final Answer and Conclusion

Therefore, $5^1 / 4 \div 2^2 / 7 = 35 / 16$, which can also be expressed as $2 \frac{3}{16}$. Comparing this result with the options given:

A) $12^{19} / 64$ B) 12 C) $219 / 64$ D) 2

None of the given options exactly match our result of $35 / 16$ or $2 \frac{3}{16}$. However, it's important to note that $219 / 64$ is not equivalent to $35 / 16$. If we simplify $219 / 64$, we get approximately 3.42, whereas $35 / 16$ is approximately 2.19. So, there appears to be no correct answer among the provided choices.

In conclusion, the correct answer to the expression $5^1 / 4 \div 2^2 / 7$ is $35 / 16$ or $2 \frac{3}{16}$. This detailed breakdown has illustrated the step-by-step process of simplifying the expression, emphasizing the importance of following the correct order of operations and understanding fraction arithmetic. While the provided options do not include the correct answer, the process of solving the problem remains valuable in enhancing mathematical skills and comprehension.

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Solving $5^1 / 4 \div 2^2 / 7$: A Step-by-Step Guide