Calculating Ms. Price's Salary: A Math Problem
Hey there, math enthusiasts! Today, we're diving into a fun little problem about Ms. Price and her salary. The core of the question lies in understanding how successive percentage increases affect a value. The problem provides us with Ms. Price's current salary and the percentage increase she received over the past two years, and our mission is to figure out what her salary was two years ago. We'll be using some fundamental math concepts to solve this, including percentages, and a bit of algebraic thinking. The goal is to accurately calculate Ms. Price's salary two years prior, rounded to the nearest cent. Let’s break it down and get started! This problem is a classic example of a reverse percentage problem, where we know the final value and the percentage increase, and we need to work backward to find the original value. This kind of problem isn't just about math; it's about understanding how things change over time and how to apply this knowledge to real-world scenarios. We'll start with the basics, define our variables, and then carefully work through the steps to find the solution. Are you ready to dive in?
Understanding the Problem
Alright, let’s get this show on the road! Before we jump into the numbers, it's super important to understand the situation. Ms. Price's salary has grown over the last two years. Each year, she got a 12% raise. Now, this year, her salary is $97,000. Our task is to calculate her initial salary from two years ago, before the raises. This isn’t a one-step calculation; we have to go backward, reversing the effect of those raises. Let’s consider this: A 12% raise means her salary increased by 12% of the previous year's salary. After the first raise, the salary is 112% of the original salary. After the second raise, the new salary is 112% of the salary after the first raise. This sets the stage for a calculation that unwinds these increases. It’s like peeling back layers to get to the core. We know the final layer (current salary), and we need to find the base layer (salary two years ago). It's a journey, not a destination, guys. Ready? Let's roll!
To make things easier, let's denote the salary two years ago as S. We know that after the first 12% raise, the salary becomes 1.12 * S*. After the second 12% raise, the salary is 1.12 * (1.12 * S*), and this final value equals $97,000. This is the key equation we will solve. Understanding this is crucial, and it's the foundation of solving the problem. The core idea is that each raise is a multiplication factor applied to the previous year’s salary. Our goal is to work backward by dividing out these factors. We’re working in reverse, like a time machine, going from the future back to the past. The math might seem complicated at first, but with a bit of focus, it will become very clear. This approach allows us to see how the percentages accumulate over time, ultimately leading us to the initial value. Now, let’s get into the specifics and solve it.
Step-by-Step Solution
Okay, buckle up, because here comes the math part! We're going to solve this step by step to keep it clear and easy. Remember, our main goal is to find S, the salary from two years ago. We've established that the current salary, $97,000, is the result of applying a 12% increase twice to the initial salary. So, we can represent this mathematically as follows: Current Salary = Initial Salary * (1 + percentage increase) * (1 + percentage increase). Let's put the numbers in place: .
To find S, we need to reverse these multiplications. This means we'll divide the current salary by 1.12 twice. First, divide $97,000 by 1.12. This gives us the salary after the first raise: $97,000 / 1.12 = $86,607.14 (approximately). Next, to find the initial salary, divide this result again by 1.12: $86,607.14 / 1.12 = $77,327.80 (approximately). Therefore, Ms. Price's salary two years ago was approximately $77,327.80. Rounding to the nearest cent, we get $77,327.80. See? Not as hard as it seemed, right? The key is to understand that the percentage increases compound, meaning each increase is based on the new, higher value. And when we’re solving for the original value, we reverse these compounded increases by dividing. Each step helps you get closer to the solution. Always take it slow, and don't rush the calculations. By working through it methodically, we are able to easily isolate the unknown and reveal the solution. Awesome!
Detailed Calculation
Let’s break down the calculations even further so that everyone understands every single step we took. First, let's clarify the percentage increases. A 12% increase means we’re adding 12% of the original salary to the original salary. The percentage, represented as a decimal, is 0.12. So, when we calculate the salary after the first raise, we multiply the original salary by 1 + 0.12, which equals 1.12. We apply this factor twice since there were two raises. Our equation becomes: . To isolate S, we perform the division. We divide both sides of the equation by 1.12 twice. First, we have , which gives us the salary from one year ago. Then, we take that result and divide it by 1.12 again. This gets us: $S = .
Let’s do the math: $97,000 / 1.12 = $86,607.14 (rounded to the nearest cent). Then, $86,607.14 / 1.12 = $77,327.80 (rounded to the nearest cent). Thus, two years ago, Ms. Price’s salary was $77,327.80. The essential step is understanding how to convert a percentage increase into a multiplication factor, which is 1 + (percentage as a decimal). This approach ensures we correctly account for the increase. Remember, the goal is always to isolate the variable we are trying to solve for, in this case, the original salary. The detailed calculation breaks the problem down into manageable chunks. Working through each step ensures we get the most accurate result, and it helps clarify the math. The more we practice these types of problems, the easier they become. Don’t you think so?
The Answer
So, after all the calculations, we have the answer! Ms. Price's salary two years ago was $77,327.80, rounded to the nearest cent. Congratulations! You've successfully worked through the problem. This solution not only answers the original question but also highlights how compound percentage increases work. Remember, the key is to understand that each percentage increase builds on the previous value, and to reverse the process, we use division. It's a practical application of math that shows how numbers can be used to understand financial changes over time. By breaking down the problem step-by-step, we were able to find the solution. Each step was a building block that led to the final answer. We saw how the original salary was affected by the two 12% raises, and then we reversed those effects to find the starting salary. We saw how to convert percentages into usable numbers in our calculations. This type of problem is relevant in many real-world situations, such as calculating investment returns or understanding inflation.
So, guys, what do you think? It's like a fun puzzle, right? The journey of solving the problem is as valuable as the answer itself. Understanding the underlying concepts will help you tackle similar problems with confidence in the future. Now go and have fun with math!