Math Check: Did Matthew Pay The Right Amount?
Let's break down Matthew's shopping trip to see if he paid the correct amount. We'll calculate the cost of the CDs, add the carrying case, figure out the sales tax, and then compare the total to what Matthew paid. Grab your calculators, guys, because we're diving into some math!
Calculating the Cost of the Compact Discs
First, let's figure out how much Matthew spent on the compact discs. He bought 4 CDs, and each cost $16.99. To find the total cost, we simply multiply the number of CDs by the price of each CD:
4 CDs * $16.99/CD = $67.96
So, Matthew spent $67.96 on the compact discs. This is a crucial first step because it forms the base of our calculation. We need this number to figure out the total cost before tax. Accuracy here is key to getting the correct final answer. Think of it like building a house; a shaky foundation means the whole structure is unstable. In our case, an incorrect CD cost means the entire calculation will be off, and we won't be able to determine if Matthew paid the right amount.
Make sure when you're doing these kinds of problems, you double-check your multiplication. A simple mistake can throw everything off. Also, keep an eye on the units. We're multiplying CDs by dollars per CD, which gives us a total cost in dollars – makes sense, right? This step is all about laying the groundwork for the rest of the problem. We've got the CD cost sorted, so now we can move on to the next piece of the puzzle: the carrying case. Remember, we're trying to find out the total cost of all the items before tax, so we need to include everything Matthew bought.
Next up, we'll add in the cost of that carrying case. Stay tuned!
Adding the Cost of the Carrying Case
Now that we know the cost of the CDs, let's add in the price of the carrying case. Matthew bought a carrying case for $35.89. To find the total cost of his purchases before tax, we add the cost of the CDs ($67.96) to the cost of the carrying case:
$67.96 (CDs) + $35.89 (Carrying Case) = $103.85
So, the total cost of Matthew's purchases before sales tax is $103.85. This is an important milestone in our calculation. We now know how much Matthew's items cost before the taxman gets involved. This pre-tax total is what the sales tax will be calculated on, so it's vital that this number is correct. Any error here will ripple through the rest of the calculations, affecting the final result.
Think of it like this: if you're baking a cake and you mess up the measurements of the ingredients, the final cake won't turn out right. Similarly, if we have the wrong pre-tax total, the calculated sales tax and the final amount will be incorrect. So, double-check your addition here! Make sure you've lined up the decimal points correctly and that you've carried over any necessary numbers. A little attention to detail can save you from a mathematical mishap.
We're getting closer to figuring out if Matthew paid the right amount. We know the pre-tax total, so the next step is to calculate the sales tax. Once we have that, we can add it to the pre-tax total to find the grand total. Keep your calculators handy; we're not done yet!
Next, we'll tackle that tricky sales tax calculation.
Calculating the Sales Tax
Alright, let's figure out the sales tax. Matthew paid an 8 1/4% sales tax. First, we need to convert this percentage into a decimal. Remember that 8 1/4% is the same as 8.25%. To convert a percentage to a decimal, we divide by 100:
8.25% / 100 = 0.0825
So, the sales tax rate as a decimal is 0.0825. Now, we multiply this decimal by the total cost of Matthew's purchases before tax ($103.85) to find the amount of sales tax:
$103.85 * 0.0825 = $8.562375
Since we're dealing with money, we need to round this to the nearest cent. $8.562375 rounds to $8.56. Therefore, the sales tax on Matthew's purchases is $8.56.
Sales tax can be a bit confusing, but it's essential to get it right. Remember, the sales tax is a percentage of the purchase price that's added to the total cost. It's a way for the government to collect revenue. In this case, Matthew has to pay an additional $8.56 on top of the cost of the CDs and the carrying case.
Make sure you understand how to convert percentages to decimals and how to multiply decimals. These are fundamental skills that are useful in many real-life situations, not just in math problems. Also, always remember to round to the nearest cent when dealing with money. It's a small detail, but it can make a difference in the final answer.
Now that we've calculated the sales tax, we're just one step away from finding out if Matthew paid the right amount. We need to add the sales tax to the pre-tax total to find the grand total. Let's do that next!
Determining the Total Amount Due
Okay, we're in the home stretch! To find the total amount Matthew should have paid, we need to add the sales tax ($8.56) to the total cost of his purchases before tax ($103.85):
$103.85 + $8.56 = $112.41
So, the total amount Matthew should have paid is $112.41. Now, let's compare this to the amount Matthew actually paid, which was $112.42.
It's super important to double-check all your calculations before making a final conclusion. Small errors can creep in, especially when dealing with multiple steps. Ensure that you've correctly added the pre-tax total and the sales tax. A tiny mistake in addition can lead to an incorrect final answer, which would then lead us to the wrong conclusion about whether Matthew paid the right amount.
Remember, precision is key in mathematics, especially when dealing with money. Always take your time and double-check your work. It's better to spend a few extra seconds verifying your calculations than to rush through and make a mistake.
We're almost there! We have the total amount Matthew should have paid and the amount he actually paid. Now, let's compare these two numbers to see if there's a difference.
Comparing the Calculated Total with the Amount Paid
We calculated that Matthew should have paid $112.41, but he actually paid $112.42. Let's find the difference between these two amounts:
$112.42 (Amount Paid) - $112.41 (Calculated Total) = $0.01
The difference is $0.01. This means Matthew paid one cent more than he should have. Therefore, the statement "Matthew paid $0.15 too little for his purchases" is incorrect.
It is essential to understand the significance of the difference. In this case, the difference is very small, only one cent. This could be due to rounding differences or a minor error in the original problem statement. However, it's important to note that even small differences can be significant in other contexts, such as in accounting or finance.
Also, remember that it's always a good idea to check your work thoroughly. In this case, we've gone through each step of the calculation carefully, so we can be confident in our answer. However, in more complex problems, it's often helpful to have someone else review your work to catch any errors you may have missed.
So, what's the final verdict? Did Matthew pay the right amount?
Conclusion: Did Matthew Pay the Correct Amount?
After carefully calculating the cost of the CDs, the carrying case, and the sales tax, we found that Matthew should have paid $112.41. He actually paid $112.42, which is one cent more than the calculated amount. Therefore, the statement "Matthew paid $0.15 too little for his purchases" is incorrect. He actually overpaid by one cent.
While the difference is minimal, it's important to go through the entire process to accurately determine if the amount paid was correct. These types of problems highlight the importance of precision in mathematical calculations and the need to pay attention to detail.
So, there you have it! We've solved the mystery of Matthew's shopping trip. Remember to always double-check your work and pay attention to the details. Happy calculating, guys!
