Solving Mixed Fraction Math Problem Step-by-Step

Hey math enthusiasts! Today, we're diving deep into a mixed fraction math problem, specifically: (2 4/5 + 2 2/3) : (10 13/30 - 3.6) * 1.25. Don't worry if it looks a bit intimidating at first – we'll break it down step-by-step to make sure you understand every single move. This isn't just about getting the right answer; it's about building your math skills and boosting your confidence. So, grab your pencils, and let's get started. We will start with a comprehensive explanation of how to solve the problem, and then we'll break down each step to make the process easier. The goal is to make sure you can tackle similar problems with ease. This problem involves adding, subtracting, and dividing fractions and mixed numbers, along with a decimal, so it covers several key areas of mathematics. This approach not only provides the solution but also builds a strong foundation for future mathematical challenges. We aim to equip you with the knowledge and confidence to handle various types of fraction problems. Remember, practice is key! The more you work through these examples, the more comfortable and proficient you'll become. So, without further ado, let's jump right into the solution!

To begin, let's understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This tells us the sequence in which to solve the equation. First, we tackle the operations inside the parentheses. In the first set of parentheses, we have the addition of mixed fractions (2 4/5 + 2 2/3). In the second set, we subtract a decimal from a mixed number (10 13/30 - 3.6). After solving these, we perform the division operation indicated by the colon (:), and finally, we multiply by 1.25. Each step will be clearly outlined to ensure clarity.

Step 1: Solving the First Parentheses – Adding Mixed Fractions

Alright, first things first, let's tackle the addition of mixed fractions: (2 4/5 + 2 2/3). To add mixed fractions, the best approach is to convert them into improper fractions. An improper fraction is simply a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This makes the addition and subtraction much easier. Let's convert our first mixed fraction, 2 4/5. To do this, we multiply the whole number (2) by the denominator (5) and then add the numerator (4). So, (2 * 5) + 4 = 14. This becomes the new numerator, and we keep the same denominator, 5. Therefore, 2 4/5 becomes 14/5.

Now, let's do the same for the second mixed fraction, 2 2/3. Multiply the whole number (2) by the denominator (3), and add the numerator (2). Thus, (2 * 3) + 2 = 8. So, 2 2/3 converts to 8/3. Now, we have 14/5 + 8/3. To add these, we need a common denominator, which is the least common multiple (LCM) of 5 and 3. The LCM of 5 and 3 is 15. We convert each fraction to have a denominator of 15. To change 14/5, multiply both the numerator and denominator by 3: (14 * 3) / (5 * 3) = 42/15. For 8/3, multiply both by 5: (8 * 5) / (3 * 5) = 40/15. Now we have 42/15 + 40/15, add the numerators: 42 + 40 = 82. So, the result of the first parentheses is 82/15.

Step 2: Solving the Second Parentheses – Subtracting Decimal from Mixed Fraction

Next, we'll solve the operations inside the second parentheses: (10 13/30 - 3.6). Before we can subtract, let's convert the mixed fraction 10 13/30 into an improper fraction, like we did before. Multiply the whole number (10) by the denominator (30) and add the numerator (13), that is (10 * 30) + 13 = 313. Thus, the mixed fraction 10 13/30 becomes 313/30. Now, we have to subtract 3.6 from 313/30. Since we have a decimal, it's best to convert it to a fraction. The decimal 3.6 can be written as 36/10. However, to make the calculation easier, let's simplify the fraction by dividing both the numerator and denominator by 2. That gets us 18/5.

To subtract 18/5 from 313/30, we must find a common denominator. The least common multiple (LCM) of 30 and 5 is 30. We already have 313/30. For 18/5, we multiply the numerator and denominator by 6: (18 * 6) / (5 * 6) = 108/30. Now we can subtract: 313/30 - 108/30 = (313 - 108) / 30 = 205/30. So, the result of the second parentheses is 205/30. Therefore, the original problem is now represented as (82/15) : (205/30) * 1.25. The problem is now greatly simplified, and all the terms are in a format conducive to performing the remaining operations. This makes the next steps more straightforward and less prone to errors.

Step 3: Performing the Division

Now, for the division part. We have (82/15) : (205/30). Dividing fractions is the same as multiplying by the reciprocal of the second fraction. The reciprocal of a fraction is simply flipping the numerator and denominator. So, the reciprocal of 205/30 is 30/205. Now, our expression becomes (82/15) * (30/205). Multiply the numerators: 82 * 30 = 2460. Then, multiply the denominators: 15 * 205 = 3075. So, we now have 2460/3075. This fraction can be simplified. Both numbers are divisible by 15. Dividing both the numerator and denominator by 15, 2460 / 15 = 164 and 3075 / 15 = 205. Thus, 2460/3075 becomes 164/205. We can further simplify by dividing both by 41, which yields us 4/5. Now we have 4/5 as the result of the division operation.

Step 4: Multiplying by 1.25

Finally, we multiply the result from the division (4/5) by 1.25. But first, let’s convert 1.25 to a fraction. 1.25 can be written as 125/100, which simplifies to 5/4. So, we now need to compute (4/5) * (5/4). Multiply the numerators: 4 * 5 = 20. Then, multiply the denominators: 5 * 4 = 20. We now have 20/20, which simplifies to 1. Therefore, the answer to our original problem (2 4/5 + 2 2/3) : (10 13/30 - 3.6) * 1.25 is 1.

To recap: We started with a complex mixed fraction math problem, broke it down into smaller, manageable steps, and solved each part methodically. We converted mixed fractions to improper fractions, found common denominators, performed addition, subtraction, division, and multiplication, and simplified the results at each stage. Remember, the key is to take it one step at a time, and always double-check your work. Practice makes perfect, and with each problem you solve, you'll become more confident and proficient in math! Understanding these processes helps build a strong foundation for more complex mathematical concepts.

Conclusion: Your Fraction Math Skills Just Got a Boost!

Congratulations, guys! You've successfully navigated a mixed fraction math problem, and you've learned a ton along the way. Remember, the journey from confusion to understanding is the most rewarding part of learning. By following these steps and practicing more problems, you'll not only solve similar equations but also build a solid mathematical foundation. Feel free to revisit these steps anytime you face a similar problem. Keep practicing, keep learning, and most importantly, keep enjoying the world of math! Keep in mind that consistent practice is the most effective method for improving your mathematical skills. Every problem solved, every step followed, brings you closer to mastery. So, take on new challenges, embrace the process, and watch your math skills grow. You got this!