Calculate Total School Days: Sony's Attendance Problem
Hey guys! Let's break down this math problem about Sony's school attendance. It's a classic percentage problem, and we'll go through it together step by step. If you're scratching your head about how to figure out the total number of school days given an attendance percentage, you're in the right place. We'll make sure it all clicks by the end of this explanation. So, let's dive in and get those math gears turning!
Understanding the Problem
The core of this problem lies in understanding percentages. Percentages are just a way of expressing a number as a fraction of 100. In this case, Sony's 80% attendance means she was present for 80 out of every 100 possible school days. Our mission is to figure out what the "100" (the total number of school days) actually represents, given that her 216 days of attendance equate to that 80%. This requires us to translate the word problem into a mathematical equation, which is a fundamental skill in problem-solving. Before we jump into the solution, let's reiterate the givens:
- Sony attended school for 216 days. This is a crucial piece of information, as it represents a specific quantity.
- Her attendance rate was 80%. This percentage links her actual attendance to the total possible days.
The problem asks us to find the total number of days the school was open. This is our unknown, the value we need to calculate.
To make it even clearer, think of it like this: if the school was open for 100 days, Sony would have attended 80 of those days (80% of 100 is 80). But we know she attended 216 days, so the total number of school days must be higher. The key is setting up the correct equation to reflect this relationship.
Setting Up the Equation
The most important step in solving any word problem is translating it into a mathematical equation. This equation will act as our roadmap, guiding us to the solution. Let's break down how to do this for Sony's attendance problem.
We know that 80% of the total number of school days is equal to 216 days. We can express this as:
80% of (Total School Days) = 216 days
Now, let's replace "Total School Days" with a variable. A common choice is 'x', so our equation becomes:
80% of x = 216
But to work with percentages in an equation, we need to convert them to decimals. To do this, we divide the percentage by 100:
80% = 80 / 100 = 0.8
So, our equation now looks like this:
- 8 * x = 216
This equation is the heart of the solution. It succinctly captures the relationship between Sony's attendance, the attendance rate, and the total number of school days. The next step is to solve for 'x', which will give us the answer we're looking for. Remember, 'x' represents the total number of days the school was open, and we're on the verge of finding it!
Solving for the Unknown
Now that we've got our equation set up (0.8 * x = 216), the next step is to isolate 'x' and find its value. This involves using basic algebraic principles to manipulate the equation. Our goal is to get 'x' by itself on one side of the equation, which will tell us the total number of school days.
Currently, 'x' is being multiplied by 0.8. To undo this multiplication, we need to perform the opposite operation: division. We'll divide both sides of the equation by 0.8. This is a crucial step because it maintains the balance of the equation – what we do to one side, we must do to the other.
So, let's divide both sides by 0.8:
(0.8 * x) / 0.8 = 216 / 0.8
On the left side, the 0.8 in the numerator and the 0.8 in the denominator cancel each other out, leaving us with just 'x':
x = 216 / 0.8
Now, we just need to perform the division. You can use a calculator, long division, or whatever method you're comfortable with. 216 divided by 0.8 equals 270.
x = 270
And there we have it! We've solved for 'x', which represents the total number of school days. So, the school was open for 270 days in the year.
Verifying the Solution
It's always a good idea to double-check your answer, especially in math problems. This helps ensure you haven't made any mistakes along the way. In this case, we can verify our solution by plugging the value we found for 'x' (270 days) back into the original equation and seeing if it holds true.
Our original equation was:
80% of x = 216
Now, let's substitute 'x' with 270:
80% of 270 = 216
To calculate 80% of 270, we multiply 270 by 0.8 (remember, 80% is equal to 0.8 as a decimal):
- 8 * 270 = 216
When we perform the multiplication, we get 216, which is exactly what we expected! This confirms that our solution is correct. 80% of 270 days is indeed 216 days, which means the school was open for a total of 270 days. Verifying the solution is not just about confirming the answer; it's about reinforcing your understanding of the problem and the steps you took to solve it. It's a great habit to develop for all types of problem-solving.
Expressing the Answer Clearly
We've done the math, we've verified our solution, and now it's time to clearly state the answer. This is an important step because it ensures that anyone reading your solution can easily understand the result. In the context of this problem, simply stating "270" isn't enough. We need to provide the units and context to make the answer meaningful.
So, the answer to the question "How many days was the school open in total?" is:
The school was open for 270 days.
This answer is clear, concise, and directly addresses the question posed in the problem. It leaves no room for ambiguity and demonstrates a complete understanding of the problem and its solution. Always remember to include the appropriate units in your answer (days, in this case) to provide a full and accurate solution.
Key Takeaways
Let's recap the main things we've learned from solving this problem. This will help solidify your understanding and make you a better problem-solver in the future:
- Understanding Percentages: This problem hinges on the concept of percentages. Remember that a percentage is just a way of expressing a part of a whole as a fraction of 100. 80% means 80 out of every 100.
- Translating Words into Equations: A crucial skill in math is turning word problems into mathematical equations. We did this by representing the unknown (total school days) with a variable ('x') and expressing the relationship between Sony's attendance, the attendance rate, and the total days in an equation.
- Solving for the Unknown: We used basic algebra to isolate the variable 'x' and solve for its value. This involved performing the inverse operation (division) to undo the multiplication.
- Verifying the Solution: Always double-check your answer! We verified our solution by plugging it back into the original equation and confirming that it held true.
- Clear Communication: Express your answer clearly, including the appropriate units and context.
By mastering these key takeaways, you'll be well-equipped to tackle similar percentage problems and other mathematical challenges. Remember, practice makes perfect, so keep honing your skills!
Practice Problems
Want to put your newfound skills to the test? Here are a couple of practice problems similar to Sony's attendance question. Try solving them on your own, following the steps we've outlined above. Don't be afraid to revisit the explanation if you get stuck. The key is to practice and build your confidence.
Practice Problem 1:
- A store is having a 20% off sale on all items. If a shirt is on sale for $24, what was the original price of the shirt?
Practice Problem 2:
- A student scored 75% on a test with 80 questions. How many questions did the student answer correctly?
Work through these problems, and you'll become even more comfortable with percentages and problem-solving strategies. Remember, math is like a muscle – the more you exercise it, the stronger it gets!
Conclusion
So, there you have it! We've successfully solved Sony's attendance problem and learned some valuable problem-solving techniques along the way. From understanding percentages to setting up equations, solving for unknowns, verifying solutions, and communicating clearly, we've covered a lot of ground. Remember, math isn't just about numbers; it's about logical thinking and problem-solving skills that can be applied in many areas of life.
I hope this explanation has been helpful and has made the concept of percentage problems a little less daunting. Keep practicing, keep asking questions, and most importantly, keep believing in yourself. You've got this!