Unlocking Math Problems: Solving For A+b And Arithmetic Series
Hey everyone, let's dive into some cool math problems! We're gonna break down how to solve for a + b in a few different scenarios and then tackle an arithmetic series. Don't worry, it's not as scary as it sounds! We'll go step-by-step, making sure everything is super clear and easy to follow. Ready? Let's jump in!
1. Calculating a + b When x = 14
Alright guys, the first set of problems gives us a value for x (which is 14) and then asks us to find a + b in a few different equations. This is all about using what we know to solve for what we don't. Think of it like a puzzle where we're given some pieces and need to fit them together to find the missing one. Let's tackle each equation one by one, making sure to show every step so that you can follow along easily. We'll use basic algebra principles to isolate a + b and find its value. Remember, the goal is always to get a + b by itself on one side of the equation. So let's start with the first problem:
a) a + b - 3x = 56
Okay, so we know x = 14. Let's substitute that into the equation: a + b - 3(14) = 56. Now, let's simplify: a + b - 42 = 56. To get a + b by itself, we need to add 42 to both sides of the equation. This gives us: a + b = 56 + 42. Finally, adding those numbers together, we get a + b = 98. See? Not too bad, right? We just needed to substitute, simplify, and isolate the a + b term.
c) 7 * a + 7 * b + 7 * x = 700
Next up, we have 7a + 7b + 7x = 700. Remember that x = 14. Let's plug that in: 7a + 7b + 7(14) = 700. Simplifying, we get: 7a + 7b + 98 = 700. Now, let's subtract 98 from both sides: 7a + 7b = 700 - 98. This simplifies to: 7a + 7b = 602. Notice that both terms on the left side have a common factor of 7. We can factor that out: 7(a + b) = 602. To isolate a + b, we divide both sides by 7: (a + b) = 602 / 7. Therefore, a + b = 86. We're getting the hang of this, right? It's all about following the steps systematically.
e) 39 * a + 39 * b - 39 * x = 3900
Here, we have 39a + 39b - 39x = 3900. And, x = 14. Substitute that in: 39a + 39b - 39(14) = 3900. Simplify: 39a + 39b - 546 = 3900. Add 546 to both sides: 39a + 39b = 3900 + 546. This gives us: 39a + 39b = 4446. Notice the common factor of 39 again. We can factor that out: 39(a + b) = 4446. Now, to get a + b alone, divide both sides by 39: (a + b) = 4446 / 39. Thus, a + b = 114. Awesome, we're doing great. Keep up the good work!
g) x * a + x * b = 140
Finally, we have x * a + x * b = 140. We know x = 14. Substitute that in: 14a + 14b = 140. Factor out the 14: 14(a + b) = 140. Divide both sides by 14: (a + b) = 140 / 14. So, a + b = 10. See, with a little patience and a step-by-step approach, we've solved for a + b in all four equations. Nicely done, guys!
2. Arithmetic Series: Diving into the Sum
Alright, now let's move on to arithmetic series. This part is about finding the sum of a sequence of numbers that follow a specific pattern. It's super useful in all sorts of real-world scenarios, from calculating the total cost of something with a fixed increase to understanding patterns in data. Let's break down how to find the sum of an arithmetic series, specifically with some provided examples. It's all about identifying the pattern, using the right formula, and then doing some simple arithmetic. We'll use the principles of arithmetic series to efficiently calculate these sums. Let's start with the first series:
a) (12 + 24 + 36 + ... + 444) : (6 + 12 + 18 + ... + 66)
In this problem, we need to find the sum of two arithmetic series and then divide the results. First, let's look at the numerator: 12 + 24 + 36 + ... + 444. This is an arithmetic series where each term increases by 12. To find the sum, we can use the formula S = n/2 * (first term + last term), where S is the sum, and n is the number of terms. First, we need to figure out how many terms are in this series. We can do that by noticing that each term is a multiple of 12. Divide the first and last terms by 12 to find the range of multiples: 12/12 = 1 and 444/12 = 37. So, there are 37 terms. The first term is 12, and the last term is 444. Therefore, the sum of the numerator is (37/2) * (12 + 444) = (37/2) * 456 = 8436.
Now, let's look at the denominator: 6 + 12 + 18 + ... + 66. This is also an arithmetic series where each term increases by 6. Again, we need to find the number of terms. Divide the first and last terms by 6: 6/6 = 1 and 66/6 = 11. So, there are 11 terms. The first term is 6, and the last term is 66. The sum of the denominator is (11/2) * (6 + 66) = (11/2) * 72 = 396. Finally, we divide the sum of the numerator by the sum of the denominator: 8436 / 396 = 21.3. This division gives us our answer. We can see how this formula simplifies finding large sums, it does wonders!
b) (4 + 8 + 12 + ... + 2012)
Here we need to find the sum of the arithmetic series 4 + 8 + 12 + ... + 2012. Each term increases by 4. Let's use the sum formula again: S = n/2 * (first term + last term). First, find the number of terms. Divide the first and last terms by 4: 4/4 = 1 and 2012/4 = 503. So, there are 503 terms. The first term is 4, and the last term is 2012. Therefore, the sum is (503/2) * (4 + 2012) = (503/2) * 2016 = 507,036. Awesome! Remember, the key is identifying the pattern, finding the number of terms, and then using the sum formula. You're arithmetic series masters now!
Conclusion
Well done, everyone! We've successfully navigated some math problems, tackling both solving for a + b and diving into arithmetic series. Remember, math is all about practice and understanding the basic principles. Break down complex problems into smaller steps, and you'll be able to solve anything. Keep practicing, and you'll find that these kinds of problems become easier and more intuitive over time. Keep up the fantastic work, and happy calculating! If you want to know more about similar topics, feel free to ask!