Math Problems: Step-by-Step Solutions & Word Problem

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Hey guys! Today, we're diving into some cool math problems. We'll tackle both numerical calculations and a word problem. So, grab your pencils, and let’s get started!

10. Numerical Calculations

Let's break down these calculations step by step to make sure we get the right answers. These problems might seem a bit long, but they're just a series of smaller operations. Remember the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to make things easier!

a) (744147 : 9 + 107 * 42) * 10 – 700000

Okay, this one looks like a beast, but don't worry, we'll tame it! First, we'll focus on what's inside the parentheses. Inside the parentheses, we have both division and multiplication, so let's tackle those first:

  1. Division: 744147 : 9

    • This division is crucial. Performing this division accurately sets the stage for the rest of the calculation. If we fumble here, the whole answer goes sideways. Let's do some long division, shall we? When we divide 744147 by 9, we get 82683. This step demonstrates the importance of basic arithmetic skills and the ability to perform long division accurately. It's a fundamental building block in solving more complex problems.

  2. Multiplication: 107 * 42

    • Next up, we've got 107 multiplied by 42. Multiplication is another key operation we need to nail. It's not just about getting the right number; it's about understanding the process of multiplication. So, when we multiply 107 by 42, we end up with 4494. This multiplication step highlights the importance of place value and how each digit contributes to the final product. Getting this right is essential for progressing further in the calculation.

Now, let's add the results inside the parentheses:

  • 82683 + 4494 = 87177

Great! We've simplified the expression inside the parentheses. Now we can move on to the next steps.

Next, we multiply by 10 and then subtract 700000:

  1. Multiply by 10: 87177 * 10 = 871770

    • Multiplying by 10 is like the easiest thing ever, right? We just slap a zero at the end! So, 87177 times 10 gives us 871770. This is a straightforward operation, but it’s a good reminder of the power of place value in our number system. Each digit shifts one place to the left, making the number ten times bigger. It's simple, but crucial for larger calculations.

  2. Subtraction: 871770 – 700000

    • Finally, we subtract 700000 from 871770, and we get 171770. Subtraction is the last step here, and it brings us to our final answer. We're taking away a large chunk (700000) from our intermediate result, leaving us with 171770. This step really underscores the relationship between addition and subtraction, and how they help us break down and solve problems.

So, the final answer for part a) is 171770. Woohoo! We did it!

b) 40927 + (50000 - 20995 : 5) – 100 * 308

Alright, let's tackle the next one. Same drill – we'll follow the order of operations step by step. Remember, PEMDAS/BODMAS is our best friend here!

First up, we'll deal with the parentheses again. Inside the parentheses, we have subtraction and division, so division goes first:

  1. Division: 20995 : 5

    • We need to divide 20995 by 5. Long division time again! When we crunch the numbers, we find that 20995 divided by 5 is 4199. This division step is all about breaking down a larger number into smaller, equal parts. It helps us see how many times 5 fits into 20995, which is essential for solving this part of the problem.

Now, we subtract this result from 50000:

  1. Subtraction: 50000 – 4199

    • Next, we subtract 4199 from 50000. This subtraction helps us find the difference between these two numbers. It's a straightforward subtraction problem, but accuracy is key. The result is 45801. This step highlights the importance of understanding place value and borrowing when necessary. It’s a fundamental skill for solving more complex problems.

Great! We've simplified the expression inside the parentheses to 45801. Now, let's move on to the multiplication outside the parentheses:

  1. Multiplication: 100 * 308

    • We multiply 100 by 308, and guess what? It's super easy! We just add two zeros to 308, giving us 30800. Multiplying by 100 is a neat trick because it reinforces our understanding of place value. It’s a quick way to scale up a number and can be a real timesaver in calculations.

Now, let's put it all together:

  1. Addition and Subtraction: 40927 + 45801 – 30800

    • Now, we add and subtract in the order they appear. First, we add 40927 and 45801, which gives us 86728. Then, we subtract 30800 from 86728. This final subtraction brings us to the solution. It showcases how we combine addition and subtraction to reach the final answer, and it reinforces the idea that math problems often require a sequence of steps to solve.

So, the final answer for part b) is 55928. Awesome job, team!

11. Word Problem: Aziza's Olympiad Prep

Let's switch gears and tackle a word problem. Word problems are like puzzles where we need to figure out what the question is asking and then use math to solve it. This one is about Aziza preparing for an Olympiad. Word problems are essential because they help us apply math to real-life situations.

Here's the problem again:

Aziza prepared for the Olympiad for three days. On the first day, she solved 42 problems, on the second day she solved 14 more problems than on the first day, and on the third day she solved twice as many problems as on the second day. How many problems did Aziza solve in total over the three days?

Let's break it down step by step:

  1. Problems solved on the first day: 42

    • The problem tells us right off the bat that Aziza solved 42 problems on the first day. This is our starting point, and we’ll use it to figure out how many problems she solved on the other days. This simple fact is crucial for setting up the rest of the solution. It’s like the foundation of a building – we need to know this to build on it!

  2. Problems solved on the second day: 42 + 14

    • The problem says Aziza solved 14 more problems on the second day than the first. So, we need to add 14 to the number of problems she solved on the first day. This step shows how we translate words into mathematical operations. When we see "more than," it often means addition. It's a key skill in tackling word problems!

    • 42 + 14 = 56 problems

  3. Problems solved on the third day: 56 * 2

    • On the third day, Aziza solved twice as many problems as she did on the second day. “Twice” means we need to multiply by 2. So, we multiply the number of problems she solved on the second day (56) by 2. This step is another example of translating words into math. “Twice” is a common word in word problems that signals multiplication, and recognizing it helps us set up the equation correctly.

    • 56 * 2 = 112 problems

Now, let's find the total number of problems solved over the three days:

  1. Total problems: 42 + 56 + 112

    • To find the total number of problems Aziza solved, we add up the problems she solved each day. This is the final step where we combine all our previous calculations to answer the main question. It's like the grand finale of our math problem-solving journey!

    • 42 + 56 + 112 = 210 problems

So, Aziza solved a total of 210 problems over the three days. Great job, Aziza!

Final Thoughts

We tackled some serious math today, guys! We crushed numerical calculations and conquered a word problem. Remember, the key is to break down problems into smaller steps and stay organized. Keep practicing, and you'll become math whizzes in no time! You got this! Remember, practice makes perfect, so keep those pencils moving and those brains working!