Solving √25/100 ÷ 25/100 + √1/2 × √1/8 A Step-by-Step Guide

In this article, we will explore a detailed, step-by-step solution to the mathematical expression √25/100 ÷ 25/100 + √1/2 × √1/8. This problem combines several fundamental mathematical operations, including square roots, division, addition, and multiplication. By breaking it down into manageable steps, we can ensure clarity and accuracy in our solution. Understanding the order of operations (PEMDAS/BODMAS) is crucial for tackling such problems effectively. This guide aims to not only provide the correct answer but also to explain the reasoning behind each step, making it a valuable resource for students and anyone looking to refresh their mathematical skills. So, let’s dive in and unravel this mathematical puzzle together.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin, it’s essential to understand the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations to arrive at the correct answer. Ignoring this order can lead to incorrect solutions. In our problem, we have square roots, division, addition, and multiplication. According to PEMDAS/BODMAS, we should first address any exponents (which include square roots), then multiplication and division (from left to right), and finally addition and subtraction. This systematic approach ensures we handle each part of the expression in the correct sequence, leading us to the accurate final result. By keeping this order in mind, we can methodically solve the problem, breaking it down into smaller, more manageable steps. Now, let's proceed with the actual calculations.

Step 1: Simplifying the Square Roots

The first step in solving the expression √25/100 ÷ 25/100 + √1/2 × √1/8 is to simplify the square roots. We have two square root terms: √25/100 and √1/2 × √1/8. Let's simplify √25/100 first. The square root of 25 is 5, and the square root of 100 is 10. Therefore, √25/100 simplifies to 5/10, which can be further reduced to 1/2. Next, we address the second square root component, √1/2 × √1/8. To simplify this, we first multiply the fractions inside the square roots: 1/2 multiplied by 1/8 equals 1/16. So, we now have √1/16. The square root of 1 is 1, and the square root of 16 is 4. Thus, √1/16 simplifies to 1/4. By simplifying the square roots first, we make the subsequent calculations easier and less prone to error. Now that we've handled the square roots, the expression looks much simpler and we can move on to the next set of operations.

Step 2: Performing the Division

After simplifying the square roots, our expression now looks like this: 1/2 ÷ 25/100 + 1/4. The next operation according to PEMDAS/BODMAS is division. We need to divide 1/2 by 25/100. To divide fractions, we multiply by the reciprocal of the divisor. The reciprocal of 25/100 is 100/25. Therefore, we have 1/2 multiplied by 100/25. Before performing the multiplication, we can simplify the fractions. 25 goes into 100 four times, so 100/25 simplifies to 4. Now our division becomes 1/2 multiplied by 4. Multiplying 1/2 by 4 gives us 4/2, which simplifies to 2. So, the result of the division 1/2 ÷ 25/100 is 2. This step is crucial as it reduces the complexity of the expression, making it easier to manage the remaining operations. With the division completed, we are one step closer to the final solution. Now, let's proceed to the next operation, which is addition.

Step 3: Completing the Addition

With the division completed, our expression has been simplified to 2 + 1/4. The final operation to perform is addition. We need to add 2 and 1/4. To do this, we can convert 2 into a fraction with a denominator of 4. 2 is equivalent to 8/4. So, our addition becomes 8/4 + 1/4. Adding these fractions involves adding the numerators while keeping the denominator the same. Thus, 8/4 + 1/4 equals 9/4. This fraction is an improper fraction, meaning the numerator is greater than the denominator. We can convert it to a mixed number to make it easier to understand. 9 divided by 4 is 2 with a remainder of 1. Therefore, 9/4 is equal to 2 and 1/4. So, the final result of the addition 2 + 1/4 is 2 1/4 or 9/4. This completes the final step in solving the mathematical expression. We have now successfully navigated through all the operations, following the correct order and arriving at the final answer.

Final Answer

After following all the steps, we have arrived at the final answer to the expression √25/100 ÷ 25/100 + √1/2 × √1/8. We first simplified the square roots, then performed the division, and finally completed the addition. The final result is 2 1/4 or 9/4. This process highlights the importance of understanding the order of operations and breaking down complex problems into smaller, manageable steps. By doing so, we can ensure accuracy and clarity in our solutions. Mathematics often seems daunting, but with a systematic approach and a clear understanding of the fundamental principles, even complex expressions can be solved with confidence. This exercise not only provides the answer but also reinforces the importance of methodical problem-solving in mathematics and beyond.