Long Family's School Expenses: Did They Pay The Correct Amount?

Hey guys! Ever feel like you're double-checking every receipt to make sure you weren't overcharged? Today, we're diving into a real-life scenario faced by the Long family. They spent money on school supplies and new clothes, and we're going to figure out if the total amount they paid, including sales tax, is accurate. Math can be a bit tricky sometimes, but don't worry, we'll break it down step by step so it's super easy to follow. So, grab your calculators (or just your brains!) and let's get started!

Understanding the Long Family's Expenses

To determine if the Long family paid the correct amount, we need to understand exactly what they spent their money on. The problem tells us they spent $38.62 on school supplies and $215.78 on new school clothes. These are our two main expenses. It's important to keep these figures separate for now because we'll need the total cost of goods before we can calculate the sales tax.

But wait, what about the sales tax? Ah, that's the crucial part! They paid 6 4/5% sales tax on their purchases. This percentage is a bit unusual looking, so we'll need to convert it into a more manageable form, like a decimal, before we can use it in our calculations. Remember, percentages are just fractions out of 100, so 6 4/5% means 6 and 4/5 out of 100. We'll tackle that conversion shortly. For now, let's just keep in mind that sales tax is an additional cost on top of the price of the items they bought. Getting a handle on these initial costs is the first big step in solving the mystery of whether the Long family paid the right amount. We've got the individual costs, and we know there's a sales tax involved – now it's about putting all the pieces together!

Calculating the Total Cost Before Tax

Okay, before we can even think about sales tax, we need to figure out the total cost of the school supplies and clothes. This is a pretty straightforward step: we simply add the two amounts together. So, we're adding $38.62 (school supplies) and $215.78 (new school clothes). You can punch this into your calculator, or if you're feeling brave, do it by hand! Either way, the goal is to find the sum of these two expenses.

Adding these amounts, we get a total of $254.40. This is the amount the Long family spent before sales tax. It's a crucial number because the sales tax is calculated based on this subtotal. Think of it like this: the store charges this much for the items themselves, and then the government adds a little extra (the sales tax) on top. Now that we know the pre-tax amount, we're one step closer to figuring out the sales tax itself. We've laid the groundwork, and the next stage is all about converting that percentage and applying it to our total. It’s like building a puzzle – we’ve got a big chunk of it done already!

Converting the Sales Tax Percentage

Alright, let's tackle that tricky sales tax percentage: 6 4/5%. It looks a bit intimidating as a mixed number percentage, but we can totally handle this! The first step is to convert the mixed number (6 4/5) into an improper fraction. To do this, we multiply the whole number (6) by the denominator (5) and then add the numerator (4). This gives us (6 * 5) + 4 = 34. We then keep the same denominator, so 6 4/5 becomes 34/5.

Now we have 34/5%, which is much better, but we still need to turn it into a decimal to make calculations easier. Remember, a percentage is just a fraction out of 100, so 34/5% is the same as (34/5) / 100. To divide a fraction by a whole number, we can think of the whole number as a fraction over 1 (100/1) and then multiply by the reciprocal. So, (34/5) / (100/1) becomes (34/5) * (1/100), which equals 34/500.

Finally, to convert 34/500 to a decimal, we simply divide 34 by 500. This gives us 0.068. So, 6 4/5% is equivalent to the decimal 0.068. Phew! That was a bit of a journey, but we got there. Now we have the sales tax rate in a format we can easily use to calculate the actual tax amount.

Calculating the Sales Tax Amount

Now that we've conquered the percentage conversion, we're ready to calculate the actual amount of sales tax the Long family paid. Remember, sales tax is calculated as a percentage of the total cost before tax. We already know the total cost before tax is $254.40, and we've converted the sales tax rate to a decimal: 0.068. So, to find the sales tax amount, we simply multiply these two numbers together: $254.40 * 0.068.

When you multiply those numbers, you should get $17.2992. Now, since we're dealing with money, we need to round this to the nearest cent. The third decimal place is a 9, which is greater than or equal to 5, so we round the second decimal place up. This means the sales tax amount is approximately $17.30. We've now figured out how much extra the Long family had to pay due to sales tax. This is a major piece of the puzzle, and we’re getting closer to the final answer!

Calculating the Total Amount Paid

We're almost there! We've calculated the cost of the items before tax ($254.40) and the sales tax amount ($17.30). To find the total amount the Long family should have paid, we simply add these two figures together: $254.40 + $17.30. This will give us the grand total, including tax, that they should have seen on their receipt.

Adding these amounts, we get $271.70. This is the total amount the Long family should have paid. We've done all the hard work of breaking down the expenses, converting the percentage, calculating the sales tax, and adding it all together. Now we have a number to compare against the amount they actually paid.

Determining if the Long Family Paid the Correct Amount

Okay, the moment of truth! We've calculated that the Long family should have paid $271.70 in total. The original problem stated that they actually paid $269.07. Now we need to compare these two amounts and see if they match up. Did they pay the correct amount, or was there a mistake somewhere?

Comparing the two numbers, we see that $269.07 is less than $271.70. This means the Long family paid less than we calculated they should have. The difference isn't huge, but it's definitely there. To find the exact difference, we can subtract the amount they paid from the amount they should have paid: $271.70 - $269.07 = $2.63.

So, the Long family paid $2.63 less than what we calculated. This could be due to a variety of reasons, such as a small error in the sales tax calculation at the store, a coupon they used that wasn't mentioned in the problem, or perhaps a slight discount they received. While it's a small discrepancy, it highlights the importance of always double-checking your receipts! In conclusion, based on the information provided, it appears the Long family did not pay the exact correct amount. They were undercharged by $2.63. We did it, guys! We broke down a real-world math problem and figured out the answer. Go team!