Step-by-Step Solution (2/3 + 1/12 + 2) ÷ (1/100) - 1000
In the realm of mathematics, precision and accuracy are paramount. To achieve these, understanding the order of operations is crucial. This article delves into the step-by-step solution of the expression (2/3 + 1/12 + 2) ÷ (1/100) - 1000
, providing a comprehensive guide to mastering this fundamental concept. Whether you're a student grappling with basic arithmetic or an enthusiast seeking to refine your mathematical skills, this detailed breakdown will equip you with the knowledge and confidence to tackle similar problems effectively.
Understanding the Order of Operations (PEMDAS/BODMAS)
Before diving into the solution, let's recap the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms dictate the sequence in which mathematical operations should be performed:
- Parentheses / Brackets
- Exponents / Orders
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
By adhering to this order, we ensure consistent and accurate results in mathematical calculations. In our specific problem, (2/3 + 1/12 + 2) ÷ (1/100) - 1000
, we will first address the operations within the parentheses, followed by division, and finally subtraction. This systematic approach is the key to unlocking the correct solution.
Step-by-Step Solution
1. Solve the Parentheses: (2/3 + 1/12 + 2)
The first step is to simplify the expression within the parentheses. This involves adding the fractions and the whole number. To do this effectively, we need to find a common denominator for the fractions 2/3 and 1/12. The least common multiple (LCM) of 3 and 12 is 12. Thus, we convert 2/3 to an equivalent fraction with a denominator of 12:
(2/3) * (4/4) = 8/12
Now we can rewrite the expression within the parentheses:
8/12 + 1/12 + 2
Next, we add the fractions:
8/12 + 1/12 = 9/12
We can simplify 9/12 by dividing both the numerator and denominator by their greatest common divisor, which is 3:
9/12 = (9 ÷ 3) / (12 ÷ 3) = 3/4
Now we add the whole number 2 to the simplified fraction:
3/4 + 2
To add a fraction and a whole number, we can convert the whole number into a fraction with the same denominator. In this case, we convert 2 into a fraction with a denominator of 4:
2 = 2/1 = (2 * 4) / (1 * 4) = 8/4
Now we can add the fractions:
3/4 + 8/4 = 11/4
Therefore, the expression within the parentheses simplifies to 11/4. This completes the first crucial step in solving the problem. Remember, paying close attention to detail in fraction manipulation is key to accurate calculations.
2. Perform the Division: (11/4) ÷ (1/100)
Now that we've simplified the parentheses, the next operation according to PEMDAS/BODMAS is division. We need to divide the result from the parentheses, 11/4, by 1/100. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/100 is 100/1, which is simply 100.
So, we rewrite the division as multiplication:
(11/4) ÷ (1/100) = (11/4) * (100/1)
Now, we multiply the fractions:
(11/4) * (100/1) = (11 * 100) / (4 * 1) = 1100/4
We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4:
1100/4 = (1100 ÷ 4) / (4 ÷ 4) = 275/1 = 275
Therefore, the division results in 275. This step showcases the importance of understanding the relationship between division and multiplication of fractions. Mastering this concept is crucial for tackling more complex mathematical problems.
3. Perform the Subtraction: 275 - 1000
The final step in solving the expression is subtraction. We subtract 1000 from the result of the division, which is 275.
275 - 1000
This is a subtraction problem where we are subtracting a larger number from a smaller number. The result will be a negative number. To find the difference, we can subtract the smaller number from the larger number and then apply a negative sign:
1000 - 275 = 725
Since we are subtracting 1000 from 275, the result is negative:
275 - 1000 = -725
Therefore, the final answer to the expression (2/3 + 1/12 + 2) ÷ (1/100) - 1000
is -725. This final step underscores the importance of understanding integer operations, especially subtraction involving negative numbers. Accurate subtraction is essential for achieving the correct final result.
Conclusion
Solving the expression (2/3 + 1/12 + 2) ÷ (1/100) - 1000
demonstrates the significance of adhering to the order of operations (PEMDAS/BODMAS). By systematically addressing the parentheses, division, and subtraction, we arrived at the solution of -725. This exercise not only reinforces the importance of the order of operations but also highlights the necessity of mastering fraction manipulation and integer arithmetic. Whether you're a student learning the basics or someone looking to sharpen their math skills, the principles outlined in this guide will undoubtedly prove invaluable. Consistent practice and a thorough understanding of these fundamental concepts are the keys to mathematical proficiency. Remember, mathematics is a building block, each concept building upon the previous one, and a solid foundation in these basics will pave the way for tackling more complex challenges in the future.
This comprehensive guide serves as a valuable resource for anyone seeking to improve their mathematical abilities. By understanding and applying the principles discussed here, you can confidently approach a wide range of mathematical problems and achieve accurate results. Embrace the challenge, and continue to explore the fascinating world of mathematics!