Solving The Birthday Card And Postcard Price Puzzle

Hey guys! Ever been stuck trying to figure out how much stuff costs when you're buying a bunch of different things? This is like a real-life math problem, and we're gonna break it down. Imagine you're in a store, and you're looking at birthday cards and postcards. The birthday cards are a bit pricier, right? Let's dive into how to solve this price puzzle. It's all about setting up the right equations and using a little bit of algebra. Don't worry; it's not as scary as it sounds. We'll walk through it step by step, so you can totally ace it. Ready to become a price-figuring pro? Let's go!

Setting Up the Scenario: Understanding the Prices

Okay, so here's the deal. Birthday cards are five times more expensive than postcards. That's our first crucial piece of information. If we let the cost of a postcard be represented by the variable 'x', then the cost of a birthday card would be '5x'. Think of it like this: postcards are the base price, and birthday cards are a multiple of that price. Now, imagine you walk into a store and load up your basket. You grab three birthday cards and six postcards. When you get to the cashier, the total bill comes to R$84.00. That's the total amount you spent. Our mission is to find out the individual prices of each type of card. To do this we have to be super organized to not get confused. We're going to use some simple algebra to figure this out.

This problem involves setting up and solving an algebraic equation to find the individual costs of birthday cards and postcards. Start by defining variables for the unknowns, representing the costs. Formulate an equation that represents the total cost based on the information provided about the purchase (number of cards of each type and their relationship in price). After creating the equation, we'll solve it to determine the cost of the postcards, and then use that value to calculate the cost of the birthday cards. This approach is commonly used in various real-world situations where you need to calculate prices or quantities based on certain conditions.

Defining the Variables

Let's start by defining our variables: Let 'x' represent the cost of a postcard. Since a birthday card costs five times as much as a postcard, we'll represent the cost of a birthday card as '5x'. This means that if a postcard costs R$2, a birthday card would cost R$10. If a postcard costs R$3, a birthday card costs R$15, and so on. The variable 'x' is the key to unlocking the prices.

Setting Up the Equation

Next, we need to set up an equation to represent the total cost. You bought three birthday cards, and each cost '5x', so the total cost for the birthday cards is 3 * (5x) = 15x. You also bought six postcards, and each cost 'x', so the total cost for the postcards is 6x. The total spent was R$84.00, so our equation is: 15x + 6x = 84. This equation says that the total cost of the birthday cards plus the total cost of the postcards equals R$84.00. Now we just need to simplify and solve for x. See? Not so bad, right?

Solving the Equation: Finding the Prices

Now comes the fun part: solving the equation! We have the equation 15x + 6x = 84. The first step is to combine like terms. On the left side of the equation, we can add 15x and 6x together, which gives us 21x. So our equation simplifies to 21x = 84. To isolate 'x' and find its value, we need to divide both sides of the equation by 21. This gives us x = 84 / 21, which simplifies to x = 4. This means that each postcard costs R$4.00. Awesome! We're almost there!

Solving the equation involves combining like terms and isolating the variable to find the cost of the postcard. Understanding this process is crucial for a variety of mathematical problems, not just the one related to cards. This is like understanding the instructions to build something. The next step is to find out how much a birthday card costs. We know that the cost of a birthday card is 5x. Since we now know that x = 4, we can calculate the cost of a birthday card as 5 * 4 = 20. Therefore, each birthday card costs R$20.00. So we have successfully determined the individual costs of the cards.

Isolating the Variable

To solve for 'x', we need to get 'x' by itself on one side of the equation. We do this by dividing both sides of the equation by the number that's multiplied by 'x'. In our case, that number is 21. So, we divide both sides by 21: (21x) / 21 = 84 / 21. This simplifies to x = 4. This is an important step because it helps you to find out how much a postcard costs. Once you know the price of a postcard, you can easily calculate the price of a birthday card.

Calculating the Final Prices

Now that we know the value of x (the cost of a postcard), we can find the cost of a birthday card. Remember that a birthday card costs 5x. So, to find the price of a birthday card, we simply multiply 5 by 4 (the value of x). This gives us 5 * 4 = 20. Therefore, a birthday card costs R$20.00. Great job! We have now figured out the price of each type of card. You can use this knowledge in the future when you are out shopping and buying stuff.

Conclusion: Wrapping Up the Problem

Alright, guys, we did it! We started with a word problem and, using some basic algebra, figured out the price of postcards and birthday cards. Postcards cost R$4.00 each, and birthday cards cost R$20.00 each. This method can be applied to all sorts of real-world situations, such as figuring out the cost of different items, calculating the total cost of a meal, or even splitting expenses with friends. The key is to break down the problem into smaller parts, define variables, set up the equation, and solve it step by step.

By understanding how to set up and solve this type of problem, you're building a strong foundation for future math concepts. You're not just learning how to solve a single problem, but you're also developing critical thinking and problem-solving skills that will serve you well in all areas of your life. So pat yourselves on the back. You've successfully navigated the price puzzle and emerged victorious! Keep practicing, and you'll become a pro at solving word problems in no time. Now go and enjoy your shopping trip, armed with your new math skills.

Summary of the Findings

To recap, we found that each postcard costs R$4.00 and each birthday card costs R$20.00. These values were determined by first setting up an equation that represents the relationships between the prices and then solving the equation. We've learned that the cost of a birthday card is five times the cost of a postcard. When the cost of three birthday cards and six postcards is added up, the total amount is R$84.00. Through solving a simple algebraic equation we were able to find the price for both types of card. This method can be used to solve many other price calculation problems.

Applying the Knowledge

This method can be used when you're shopping and want to figure out the cost of multiple items. It's helpful for budget planning and managing your money. For example, you can estimate the cost of multiple items. For instance, imagine you want to buy 5 postcards and 2 birthday cards. With this knowledge, you can calculate the total cost of your purchase by multiplying the unit costs by the quantity of each item and adding the results. This way, you'll be able to do the maths in your head. The more you practice, the better you'll become at these calculations.