Calculate Carla's Savings Over 10 Weeks Math Problem Solved

Hey guys! Today, we're diving into a fun math problem that's super practical in real life: saving money! We're going to figure out how much Carla saved over 10 weeks, and it's a great example of how consistent effort can really add up. So, let's jump right in and break down the problem step by step. This is not just about getting to the right answer; it's also about understanding the process and how to apply similar strategies in your own financial life. Think of it as a mini-lesson in personal finance, disguised as a math problem! We'll explore the concept of arithmetic progression and how it can help us calculate savings, plan budgets, and even understand investments. So, grab your calculators (or just your brains!), and let's get started!

Understanding the Problem

So, here's the deal: Carla is a super-saver! She started her saving journey by putting away R$5.00 in the first week. That's a great start, right? But here's the awesome part – she decided to increase her savings by R$2.00 every single week. This means she saved R$7.00 in the second week, R$9.00 in the third week, and so on. By the tenth week, she managed to save R$23.00. That's impressive!

Our mission is to figure out the total amount Carla saved over these 10 weeks. This isn't just a simple addition problem where we add up the same amount ten times. Instead, we have a series of increasing amounts, which makes it a bit more interesting. We're essentially dealing with a sequence of numbers, where each number is R$2.00 more than the previous one. This type of sequence is known as an arithmetic progression, and there are some neat formulas we can use to solve this efficiently. But before we get to the formulas, let's think about the problem conceptually. We could, of course, manually add up each week's savings, but that could be a little time-consuming, especially if we were dealing with a larger number of weeks. So, we'll explore a more elegant and quicker solution.

Breaking Down the Savings Pattern

Let’s really understand what’s happening here. Week by week, Carla's savings are growing. We can list out her savings for each week to visualize the pattern:

• Week 1: R$5.00
• Week 2: R$7.00
• Week 3: R$9.00
• Week 4: R$11.00
• Week 5: R$13.00
• Week 6: R$15.00
• Week 7: R$17.00
• Week 8: R$19.00
• Week 9: R$21.00
• Week 10: R$23.00

See the pattern? It’s a steady increase of R$2.00 each week. This pattern is key to solving the problem efficiently. If we were to add these all up manually, we would get the correct answer, but there's a more strategic way. We're essentially dealing with an arithmetic series, which is the sum of an arithmetic progression. Think of it like climbing a staircase where each step is the same height. We know the height of the first step (the first week's savings) and how much higher each step gets (the weekly increase). Our goal is to find the total height we've climbed after 10 steps (the total savings after 10 weeks). Understanding this pattern not only helps us solve this specific problem, but it also gives us a valuable tool for tackling other financial planning scenarios. For instance, we could use this same logic to calculate the total amount saved in a retirement fund with regular contributions or the total interest earned on a loan with fixed repayments. The beauty of math is that it provides us with frameworks that can be applied to a wide range of situations.

Calculating the Total Savings

Now for the fun part – the actual calculation! There are a couple of ways we can approach this, but let's focus on the most efficient method using the formula for the sum of an arithmetic series. This formula is a real lifesaver when dealing with sequences like this, as it saves us from having to add up all the individual amounts manually. The formula is as follows:

S = (n/2) * (a + l)

Where:

• S is the sum of the series (the total savings we want to find).
• n is the number of terms (in this case, the number of weeks, which is 10).
• a is the first term (Carla's savings in the first week, R$5.00).
• l is the last term (Carla's savings in the tenth week, R$23.00).

See how each part of the formula corresponds to the information we have in the problem? This is what makes it so powerful. We're essentially taking the average of the first and last terms and multiplying it by the number of terms. This gives us the total sum of the series. Let's plug in the values and see what we get:

S = (10/2) * (5 + 23)

Applying the Formula Step-by-Step

Let's break down the calculation step-by-step to make sure we understand exactly what's happening. First, we divide the number of weeks (n = 10) by 2: 10 / 2 = 5. Next, we add the first week's savings (a = R$5.00) to the tenth week's savings (l = R$23.00): 5 + 23 = 28. Now, we multiply these two results together: 5 * 28 = 140. So, according to the formula, the total savings (S) is R$140.00. Isn't it amazing how a simple formula can give us the answer so quickly? This formula works because it takes advantage of the arithmetic pattern. By averaging the first and last terms, we're essentially finding the