Math Problems: Calculating Expression Values

Hey math whizzes! Today, we're diving into some awesome math problems that will really get your brains buzzing. We've got a couple of expression calculations to tackle, plus a fun word problem involving money. Let's break them down step-by-step and make sure we get the right answers, shall we? Get your calculators ready, or even better, let's try solving these by hand to really flex those mental muscles!

Problem 1: Expression Calculation (Part 1)

Alright guys, let's kick things off with our first expression: (629 - 259) ÷ 10 + (156 - 96) x 6. This one tests your understanding of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). We need to solve the operations within the parentheses first, then handle the division and multiplication, and finally, the addition.

First up, let's deal with the parentheses. We have two sets: (629 - 259) and (156 - 96). For the first set, 629 - 259 equals 370. For the second set, 156 - 96 equals 60. Now our expression looks like this: 370 ÷ 10 + 60 x 6. See how much simpler that is already? Just by tackling those parentheses, we've made great progress. Remember, patience and following the rules are key in math, just like in life!

Next, we move on to multiplication and division, working from left to right. We have 370 ÷ 10 and 60 x 6. Let's solve these. 370 ÷ 10 is 37. And 60 x 6 is 360. Our expression is now reduced to: 37 + 360. We're almost there, folks! This is where the final calculation happens, and it's a straightforward addition.

Finally, we perform the addition: 37 + 360. Adding these two numbers gives us a grand total of 397. So, the value of the first expression is 397. Awesome job if you followed along and got the same result! It's super satisfying when you work through a problem like this and nail the answer, right? Keep this process in mind for any complex expression you encounter. Breaking it down piece by piece is the secret sauce!

Problem 2: Expression Calculation (Part 2)

Now for our second expression, which is a bit more complex: (1050 ÷ 50 - 15) ÷ (246 - 61 x 4) + (240 - 90) ÷ 6. This problem has multiple sets of parentheses and a mix of operations, so we really need to be on our toes and stick to PEMDAS diligently. Remember, each step brings us closer to the solution!

Let's start with the first set of parentheses: (1050 ÷ 50 - 15). Inside these parentheses, we have division and subtraction. According to PEMDAS, division comes before subtraction. So, first, we calculate 1050 ÷ 50. This equals 21. Now the expression inside the parentheses becomes 21 - 15, which equals 6. So, the first part simplifies to 6.

Moving on to the second set of parentheses: (246 - 61 x 4). Here, we have subtraction and multiplication. Multiplication takes precedence. So, we calculate 61 x 4, which is 244. Now, the expression inside becomes 246 - 244, which equals 2. This second part simplifies to 2.

Next, we tackle the third set of parentheses: (240 - 90). This is a simple subtraction. 240 - 90 equals 150. This third part simplifies to 150.

Now, let's substitute these simplified values back into our original expression. It now looks like this: 6 ÷ 2 + 150 ÷ 6. We've successfully navigated all the parentheses! High fives all around!

We're now at the stage of division and addition. According to PEMDAS, we perform division before addition, working from left to right. First, we have 6 ÷ 2, which equals 3. Then, we have 150 ÷ 6, which equals 25. Our expression is now just 3 + 25.

Finally, the last step is addition: 3 + 25. This gives us a grand total of 28. So, the value of the second expression is 28. Phew! That was a journey, but we got there. It really shows how important it is to follow the order of operations step-by-step. Don't rush, and double-check each calculation. You've got this!

Problem 3: Money Matters with Nafar and Javohir

Now, let's shift gears to a real-world scenario involving money. We have two friends, Nafar and Javohir, who initially had the same amount of money. Nafar spends 7000 sums daily, while Javohir spends 5000 sums daily. The question is: After how many days will Javohir have 10,000 sums more than Nafar? This is a classic problem that involves understanding rates of change and setting up an equation.

Let's define our variables. Let 'd' represent the number of days that have passed. Since they started with the same amount of money, let's call that initial amount 'M'.

After 'd' days, the amount of money Nafar has left will be his initial amount minus his total spending. His daily spending is 7000 sums, so after 'd' days, he will have spent 7000 * d sums. Therefore, Nafar's remaining money is M - 7000d.

Similarly, after 'd' days, Javohir will have spent 5000 * d sums. Javohir's remaining money is M - 5000d.

We are looking for the number of days 'd' when Javohir will have 10,000 sums more than Nafar. This means: Javohir's money = Nafar's money + 10,000.

Let's plug in our expressions for their remaining money: (M - 5000d) = (M - 7000d) + 10,000.

Now, we need to solve this equation for 'd'. Notice that the initial amount 'M' appears on both sides of the equation. We can subtract 'M' from both sides, which effectively cancels it out. This is great because it means the initial amount of money they had doesn't actually matter for solving this specific problem!

Our equation simplifies to: -5000d = -7000d + 10,000.

Now, let's get all the terms with 'd' on one side. We can add 7000d to both sides of the equation: -5000d + 7000d = 10,000.

This simplifies to: 2000d = 10,000.

Finally, to find 'd', we divide both sides by 2000: d = 10,000 ÷ 2000.

Calculating this division, we find d = 5.

So, after 5 days, Javohir will have 10,000 sums more than Nafar. Let's do a quick check to see if this makes sense. After 5 days: Nafar spends: 5 * 7000 = 35,000 sums. Javohir spends: 5 * 5000 = 25,000 sums. Nafar spends 10,000 sums more than Javohir. Since they started with the same amount, this means Javohir will have 10,000 sums less than Nafar. Wait, the question asked when Javohir will have more. Let's re-read!

Ah, I see the confusion! My apologies, guys. Let's correct the setup. Javohir spends less money per day than Nafar. This means Javohir's money will decrease slower than Nafar's. So, if Nafar spends 7000 and Javohir spends 5000, Nafar's money is going down faster. Therefore, at some point, Javohir will have more money than Nafar if we consider the difference in their spending rates.

Let's re-evaluate the difference. Nafar spends 2000 sums more per day than Javohir (7000 - 5000 = 2000). This means the gap between Nafar's money and Javohir's money increases by 2000 each day, with Nafar's money decreasing faster. This implies Nafar will always have less money than Javohir if we look at it this way, unless the question implies something else or there was a typo.

Let's assume the question implies: 'After how many days will the difference in their money be 10,000 sums, with Javohir having more?' If Nafar spends more, Javohir's money will eventually become higher relative to Nafar's starting point because his spending is lower. The difference in their spending is 2000 sums per day. For Javohir to have 10,000 more than Nafar, the initial amount 'M' must be considered. Let's stick to the equation setup:

Javohir's Money = Nafar's Money + 10,000 M - 5000d = M - 7000d + 10,000

This led us to 2000d = 10,000, which means d = 5. Let's re-check the state after 5 days:

Nafar's money: M - 7000 * 5 = M - 35,000 Javohir's money: M - 5000 * 5 = M - 25,000

Now, let's find the difference: Javohir's money - Nafar's money. (M - 25,000) - (M - 35,000) M - 25,000 - M + 35,000 35,000 - 25,000 = 10,000.

Yes! It works out perfectly. After 5 days, Javohir will indeed have 10,000 sums more than Nafar. It's all about setting up the equation correctly and understanding what the variables represent. Even experienced mathematicians sometimes need to re-read and double-check their logic, so don't feel bad if you have to pause and think things through!

Conclusion

So there you have it, guys! We tackled some challenging math expressions using the order of operations and solved a practical money problem by setting up and solving an algebraic equation. Remember, math isn't just about numbers; it's about problem-solving, logical thinking, and developing skills that help us understand the world around us. Keep practicing, keep asking questions, and never shy away from a good math challenge! You're all doing great!