Solving Math Problems: Calculations And Fractions

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Hey guys! Let's dive into some cool math problems. We're going to calculate a few expressions and then convert some decimals into fractions. It's going to be a fun journey through numbers, so buckle up! In this article, we'll break down the steps and hopefully make everything super clear and easy to follow. Get ready to flex those math muscles and sharpen your skills. We'll be tackling calculations involving decimals, fractions, and the order of operations. This is like a mini-workout for your brain, so let's get started. These problems are designed to test your understanding of basic arithmetic operations and the ability to work with different forms of numbers. So, whether you're a math whiz or just looking to brush up on your skills, this is the perfect place to do it. We'll go through the calculations step by step, ensuring you understand each move and why we do it. No need to worry if you feel a little rusty; we'll refresh all the concepts as we go. Math is all about practice, and the more you practice, the better you get. So, grab your pencils and let's get solving!

Calculation Problems

First up, we have two calculation problems. These are great examples of how you apply the order of operations and work with fractions and decimals. Remember, the order of operations is super important (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Getting this right is key to solving the problems correctly. Let's get right into it, shall we? We are going to tackle this step by step, making sure that every piece of the puzzle fits nicely. Don't worry if it looks intimidating at first; we'll break it down.

Problem 1: Step-by-Step Calculation

Here's the first problem. Let's do this!

  1. (0,645:0,3-1(107/180))â‹…(4:6,25-1,5+(1/7)â‹…1,96)

Let's break it down bit by bit. The goal is to carefully work through the problem. This means being super careful, making no silly mistakes. Let's make sure our answer is accurate. We'll focus on the expressions inside the parentheses first, as per the order of operations. This is super important! It sets the stage for everything else. Remember, doing things step-by-step is always a good strategy. Here we go!

*   **Step 1: Inside the First Parentheses.** We need to calculate what's inside the first set of parentheses: (0,645:0,3-1(107/180)). Let's simplify this part first.
    *   **Division**: First, divide 0,645 by 0,3. Then we have 0,645 / 0,3 = 2,15.
    *   **Fraction Conversion:** Now, let's change that mixed number 1(107/180) into an improper fraction. This makes things easier to manage. So, 1(107/180) = (1 * 180 + 107) / 180 = 287/180. 
    *   **Subtraction**: Now we have 2,15 - 287/180. To do this, we need to convert 2,15 into a fraction with a denominator of 180 to do the math. 2,15 = 215/100 = 43/20 = 387/180. The subtraction 387/180 - 287/180 = 100/180, which simplifies to 5/9.

*   **Step 2: Inside the Second Parentheses.** Now we attack the second set of parentheses: (4:6,25-1,5+(1/7)â‹…1,96).
    *   **Division**: First, divide 4 by 6,25. This yields 4/6,25 = 0,64.
    *   **Multiplication**: Next, multiply (1/7) by 1,96. That's (1/7) * 1,96 = 0,28.
    *   **Subtraction and Addition:**. So we have 0,64 - 1,5 + 0,28 = -0,58.

*   **Step 3: Multiplying the Results**. So we take the result of the first set of parentheses (5/9) and multiply it by the result of the second set of parentheses (-0,58). That's (5/9) * (-0,58) = (5/9) * (-58/100) = -290/900 = -29/90.

*   **Final Answer**: Therefore, the answer to the first problem is -29/90.

Problem 2: Another Calculation

Now, let's tackle the second calculation problem and break it down to make it super clear for everyone. We can do this! We'll use the same step-by-step approach we used for the first problem.

  1. ((13/50)+2,75)⋅3,6−(0,75−(1/2))⋅1,6

As always, the order of operations is our guide here! We'll focus on the parentheses, multiplication, and addition/subtraction. Remember, this is about precision and following the rules. Let's make sure we do not make any mistakes. You know you've got this!

*   **Step 1: Inside the First Parentheses.** Let's start with ((13/50)+2,75). Convert 2,75 to fraction. So, 2,75 = 275/100 = 11/4 = 137.5/50. Then, (13/50 + 137,5/50) = 150.5/50 = 3,01.
*   **Step 2: Multiplication.** Multiply the result by 3,6. Then, 3,01 * 3,6 = 10,836.
*   **Step 3: Inside the Second Parentheses.** Now, let's focus on the second set of parentheses: (0,75-(1/2)). That means (0,75 - 0,5) = 0,25.
*   **Step 4: Multiplication.** Now, we multiply 0,25 by 1,6. So, 0,25 * 1,6 = 0,4.
*   **Step 5: Subtraction:** Finally, subtract what we got in step 4 from what we got in step 2. That's 10,836 - 0,4 = 10,436.

*   **Final Answer:** Therefore, the answer to the second problem is 10,436.

Converting Decimals to Fractions

Alright, guys, let's transition to the second part of our math adventure. We are going to convert some repeating decimals into fractions. This is a super handy skill. It helps you understand numbers in a different light. It's not as scary as it looks. The basic idea is to set up an equation, multiply, and then subtract to eliminate the repeating part. Then, you simplify the fraction. We're going to use a special method for this. Let's break it down together. Here we go!

Problem 1: 1,3(1)

Let's convert 1,3(1) to a fraction.

  1. Define x: Let x = 1,3(1) = 1,3111...

  2. Multiply to Shift the Repeating Part: Multiply both sides by 10 to shift one repeating digit to the left. 10x = 13,111...

  3. Subtract to Eliminate Repeating Part: Then, subtract the original equation (x = 1,3111...) from the new one (10x = 13,111...). So, 10x - x = 13,111... - 1,3111.... This gives us 9x = 11,8.

  4. Solve for x: Now, solve for x by dividing both sides by 9. x = 11,8 / 9. Then, change into fraction. So x = 118/10 * 1/9 = 118/90 = 59/45.

    • Final Answer: Therefore, 1,3(1) equals 59/45.

Problem 2: 2,3(2)

Let's get this one done too! Convert 2,3(2) to a fraction.

  1. Define x: Let x = 2,3(2) = 2,3222...

  2. Multiply to Shift the Repeating Part: Multiply both sides by 10 to shift one repeating digit to the left. 10x = 23,222...

  3. Subtract to Eliminate Repeating Part: Then, subtract the original equation (x = 2,3222...) from the new one (10x = 23,222...). So, 10x - x = 23,222... - 2,3222.... This gives us 9x = 20,9.

  4. Solve for x: Now, solve for x by dividing both sides by 9. x = 20,9 / 9. Then, change into fraction. So x = 209/10 * 1/9 = 209/90.

    • Final Answer: Therefore, 2,3(2) equals 209/90.

Problem 3: 0,(248)

Alright, let's keep the momentum going! Convert 0,(248) to a fraction.

  1. Define x: Let x = 0,(248) = 0,248248...

  2. Multiply to Shift the Repeating Part: The repeating part has 3 digits, so multiply both sides by 1000. 1000x = 248,248248...

  3. Subtract to Eliminate Repeating Part: Then, subtract the original equation (x = 0,248248...) from the new one (1000x = 248,248248...). So, 1000x - x = 248,248248... - 0,248248.... This gives us 999x = 248.

  4. Solve for x: Now, solve for x by dividing both sides by 999. x = 248/999.

    • Final Answer: Therefore, 0,(248) equals 248/999.

Problem 4: 0,(34)

Last one, guys! Convert 0,(34) to a fraction.

  1. Define x: Let x = 0,(34) = 0,3434...

  2. Multiply to Shift the Repeating Part: The repeating part has 2 digits, so multiply both sides by 100. 100x = 34,3434...

  3. Subtract to Eliminate Repeating Part: Then, subtract the original equation (x = 0,3434...) from the new one (100x = 34,3434...). So, 100x - x = 34,3434... - 0,3434.... This gives us 99x = 34.

  4. Solve for x: Now, solve for x by dividing both sides by 99. x = 34/99.

    • Final Answer: Therefore, 0,(34) equals 34/99.

Conclusion

And there you have it, guys! We've worked through several math problems together, focusing on calculations and converting decimals to fractions. Hopefully, this has been a helpful journey, and you now feel more confident in tackling these types of problems. Remember, practice is key, so keep at it! Feel free to revisit these examples and try similar problems on your own. Keep up the excellent work, and always remember that you can conquer any math challenge with patience and effort! If you're struggling with math, don't worry. Just keep practicing. See you next time!